Short Answer
The question sounds simple, but the answer is quite involved and would depend upon factors not identified as being present or absent in the question.
There might be isolated fact patterns in which there could be liability imposed, but usually, no economic harm that couldn't be remedied by reinstatement would occur, and fairly specific factual circumstances (including intentional or willing misconduct by the officer or director) would have to be present to impose liability at all.
In all likelihood, a court would disfavor an attempt to impose liability in these circumstances even though the relief of compelling the HOA to reinstate its registration (or to have a receiver appointed for it if the directors or officers resigned first) would be easily accomplished.
Long Answer
The requirement alluded to appears to be this Chapter, Section, and Subsection of Title XL of the Florida Statutes (emphasis added):
720.303 Association powers and duties; meetings of board; official records; budgets; financial reporting; association funds; recalls.—
(1) POWERS AND DUTIES.—An association which operates a community as
defined in s. 720.301, must be operated by an association that is a
Florida corporation. After October 1, 1995, the association must be
incorporated and the initial governing documents must be recorded in
the official records of the county in which the community is located.
An association may operate more than one community. The officers and
directors of an association have a fiduciary relationship to the
members who are served by the association. The powers and duties of
an association include those set forth in this chapter and, except as
expressly limited or restricted in this chapter, those set forth in
the governing documents. After control of the association is obtained
by members other than the developer, the association may institute,
maintain, settle, or appeal actions or hearings in its name on behalf
of all members concerning matters of common interest to the members,
including, but not limited to, the common areas; roof or structural
components of a building, or other improvements for which the
association is responsible; mechanical, electrical, or plumbing
elements serving an improvement or building for which the association
is responsible; representations of the developer pertaining to any
existing or proposed commonly used facility; and protesting ad valorem
taxes on commonly used facilities. The association may defend actions
in eminent domain or bring inverse condemnation actions. Before
commencing litigation against any party in the name of the association
involving amounts in controversy in excess of $100,000, the
association must obtain the affirmative approval of a majority of the
voting interests at a meeting of the membership at which a quorum has
been attained. This subsection does not limit any statutory or
common-law right of any individual member or class of members to bring
any action without participation by the association. A member does not
have authority to act for the association by virtue of being a member.
An association may have more than one class of members and may issue
membership certificates. An association of 15 or fewer parcel owners
may enforce only the requirements of those deed restrictions
established prior to the purchase of each parcel upon an affected
parcel owner or owners.
Liability of officers who don't actually know that an entity is administratively dissolved for the acts of the entity is not permitted.
Another section of the same title and chapter is also relevant:
720.3033 Officers and directors.—
(1)(a) Within 90 days after being elected or appointed to the board,
each director shall certify in writing to the secretary of the
association that he or she has read the association’s declaration of
covenants, articles of incorporation, bylaws, and current written
rules and policies; that he or she will work to uphold such documents
and policies to the best of his or her ability; and that he or she
will faithfully discharge his or her fiduciary responsibility to the
association’s members. Within 90 days after being elected or appointed
to the board, in lieu of such written certification, the newly elected
or appointed director may submit a certificate of having
satisfactorily completed the educational curriculum administered by a
division-approved education provider within 1 year before or 90 days
after the date of election or appointment.
(b) The written certification or educational certificate is valid for
the uninterrupted tenure of the director on the board. A director who
does not timely file the written certification or educational
certificate shall be suspended from the board until he or she complies
with the requirement. The board may temporarily fill the vacancy
during the period of suspension.
(c) The association shall retain each director’s written certification
or educational certificate for inspection by the members for 5 years
after the director’s election. However, the failure to have the
written certification or educational certificate on file does not
affect the validity of any board action.
(2) If the association enters into a contract or other transaction
with any of its directors or a corporation, firm, association that is
not an affiliated homeowners’ association, or other entity in which an
association director is also a director or officer or is financially
interested, the board must:
(a) Comply with the requirements of s. 617.0832.
(b) Enter the disclosures required by s. 617.0832 into the written
minutes of the meeting.
(c) Approve the contract or other transaction by an affirmative vote
of two-thirds of the directors present.
(d) At the next regular or special meeting of the members, disclose
the existence of the contract or other transaction to the members.
Upon motion of any member, the contract or transaction shall be
brought up for a vote and may be canceled by a majority vote of the
members present. If the members cancel the contract, the association
is only liable for the reasonable value of goods and services provided
up to the time of cancellation and is not liable for any termination
fee, liquidated damages, or other penalty for such cancellation.
(3) An officer, director, or manager may not solicit, offer to accept,
or accept any good or service of value for which consideration has not
been provided for his or her benefit or for the benefit of a member of
his or her immediate family from any person providing or proposing to
provide goods or services to the association. If the board finds that
an officer or director has violated this subsection, the board shall
immediately remove the officer or director from office. The vacancy
shall be filled according to law until the end of the director’s term
of office. However, an officer, director, or manager may accept food
to be consumed at a business meeting with a value of less than $25 per
individual or a service or good received in connection with trade
fairs or education programs.
(4) A director or officer charged by information or indictment with a
felony theft or embezzlement offense involving the association’s funds
or property is removed from office. The board shall fill the vacancy
according to general law until the end of the period of the suspension
or the end of the director’s term of office, whichever occurs first.
However, if the charges are resolved without a finding of guilt or
without acceptance of a plea of guilty or nolo contendere, the
director or officer shall be reinstated for any remainder of his or
her term of office. A member who has such criminal charges pending may
not be appointed or elected to a position as a director or officer.
(5) The association shall maintain insurance or a fidelity bond for
all persons who control or disburse funds of the association. The
insurance policy or fidelity bond must cover the maximum funds that
will be in the custody of the association or its management agent at
any one time. As used in this subsection, the term “persons who
control or disburse funds of the association” includes, but is not
limited to, persons authorized to sign checks on behalf of the
association, and the president, secretary, and treasurer of the
association. The association shall bear the cost of any insurance or
bond. If annually approved by a majority of the voting interests
present at a properly called meeting of the association, an
association may waive the requirement of obtaining an insurance policy
or fidelity bond for all persons who control or disburse funds of the
association.
Florida non-profit corporations are governed by Chapter 617 of Title XXXVI of the Florida Statutes. Homeowner's Associations are taxed under Section 528 of the Internal Revenue Code (i.e. 26 U.S.C. § 528), so normally, directors and officers of an HOA are not entitled to Florida's very broad exemption from civil liability for officers and directors of non-profits that are charities. Fl. Stat. 617.0834. Therefore, the unmodified standard of liability for directors and officers apply.
The standards of conduct for other non-profit directors in Florida are as follows:
617.0830 General standards for directors.—
(1) A director shall discharge his or her duties as a director,
including his or her duties as a member of a committee:
(a) In good faith;
(b) With the care an ordinarily prudent person in a like position
would exercise under similar circumstances; and
(c) In a manner he or she reasonably believes to be in the best
interests of the corporation.
(2) In discharging his or her duties, a director may rely on
information, opinions, reports, or statements, including financial
statements and other financial data, if prepared or presented by:
(a) One or more officers or employees of the corporation whom the
director reasonably believes to be reliable and competent in the
matters presented;
(b) Legal counsel, public accountants, or other persons as to matters
the director reasonably believes are within the persons’ professional
or expert competence; or
(c) A committee of the board of directors of which he or she is not a
member if the director reasonably believes the committee merits
confidence.
(3) A director is not acting in good faith if he or she has knowledge
concerning the matter in question that makes reliance otherwise
permitted by subsection (2) unwarranted.
(4) A director is not liable for any action taken as a director, or
any failure to take any action, if he or she performed the duties of
his or her office in compliance with this section.
Also relevant (in pertinent part) is this section:
617.0831 Indemnification and liability of officers, directors, employees, and agents.— . . . . ss. 607.0831 and 607.0850 apply to a
corporation organized under this act. . . . Any reference to
“directors” in those sections includes the directors, managers, or
trustees of a corporation organized under this act. . . . However, the
term “director” as used in ss. 607.0831 and 607.0850 does not include
a director appointed by the developer to the board of directors of a
condominium association under chapter 718, a cooperative association
under chapter 719, a homeowners’ association defined in s. 720.301, or
a timeshare managing entity under chapter 721. Any reference to
“shareholders” in those sections includes members of a corporation
organized under this act[.]
Any member of the association would have standing to bring suit with an award of attorney fees if that person prevailed, but an obligation to pay attorney fees if the person bringing the suit loses. This section, however, imposes additional limitations on suits against officers and directors limiting them to "Any director or officer of an association who willfully and knowingly fails to comply with these provisions", rather than to anyone who has violated duties established elsewhere.
A dissolved non-profit corporation in Florida still exists as an entity, but is limited in the scope of what it is allowed to do:
617.1405 Effect of dissolution.—
(1) A dissolved corporation continues its corporate existence but may
not conduct its affairs except to the extent appropriate to wind up
and liquidate its affairs, including:
(a) Collecting its assets;
(b) Disposing of its properties that will not be distributed in kind
pursuant to the plan of distribution of assets adopted under s.
617.1406;
(c) Discharging or making provision for discharging its liabilities;
(d) Distributing its remaining property in accordance with the plan of
distribution of assets adopted under s. 617.1406; and
(e) Doing every other act necessary to wind up and liquidate its
affairs.
(2) Dissolution of a corporation does not:
(a) Transfer title to the corporation’s property;
(b) Subject its directors or officers to standards of conduct
different from those which applied prior to dissolution;
(c) Change quorum or voting requirements for its board of directors or
members, change provisions for selection, resignation, or removal of
its directors or officers or both, or change provisions for amending
its bylaws;
(d) Prevent commencement of a proceeding by or against the corporation
in its corporate name;
(e) Abate or suspend a proceeding pending by or against the
corporation on the effective date of dissolution; or
(f) Terminate the authority of the registered agent of the
corporation.
(3) The directors, officers, and agents of a corporation dissolved
pursuant to s. 617.1403 shall not incur any personal liability thereby
by reason of their status as directors, officers, and agents of a
dissolved corporation, as distinguished from a corporation which is
not dissolved.
(4) The name of a dissolved corporation is not available for
assumption or use by another corporation until 120 days after the
effective date of dissolution unless the dissolved corporation
provides the department with an affidavit, executed pursuant to s.
617.01201, authorizing the immediate assumption or use of the name by another corporation.
Indeed, it isn't entirely obvious that letting a Secretary of State registration lapse would actually violate Fl. Stat. § 720.303 set forth above, because the HAO would still be a Florida non-profit corporation, even if its registration has lapsed, under the statute above related to the effect of dissolution.
It is also relatively easy to reinstate a dissolved corporation:
617.1404 Revocation of dissolution.—
(1) A corporation may revoke its dissolution at any time prior to the
expiration of 120 days following the effective date of the articles of
dissolution.
(2) Revocation of dissolution must be authorized in the same manner as
the dissolution was authorized unless that authorization permitted
revocation by action of the board of directors alone, in which event
the board of directors may revoke the dissolution without member
action.
(3) After the revocation of dissolution is authorized, the corporation
may revoke the dissolution by delivering to the Department of State
for filing articles of revocation of dissolution, together with a copy
of its articles of dissolution, that set forth:
(a) The name of the corporation;
(b) The effective date of the dissolution that was revoked;
(c) The date that the revocation of dissolution was authorized;
(d) If the corporation’s board of directors revoked a dissolution
authorized by the members, a statement that revocation was permitted
by action by the board of directors alone pursuant to that
authorization; and
(e) If member action was required to revoke the dissolution, the
information required by s. 617.1403(1)(b) or (c), whichever is
applicable.
(4) Revocation of dissolution is effective upon the effective date of
the articles of revocation of dissolution.
(5) When the revocation of dissolution is effective, it relates back
to and takes effect as of the effective date of the dissolution and
the corporation resumes conducting its affairs as if dissolution had
never occurred.
Reinstatement of an administrative dissolution for not keeping current with the Secretary of State is even easier.
So, while there would clearly be a right to compel an HOA to reinstate its state registration, it isn't at all obvious what harm would arise from allowing this to lapse that couldn't be cured, so it is hard to see how a member could be exposed to money damages for not doing so. The statute generally expresses an intent not to impose liability on officers and directors of dissolved entities.
Also, there can be no liability unless the director or officer was acting in bad faith, in a manner believed by that person to be contrary to the interests of the association, or imprudently, when there is also intentional or willful misconduct by the officer or director. So merely screwing up can't give rise to liability.
I could imagine facts under which there might be liability, but they would be quite esoteric, both in terms of a fact pattern that causes economic harm, and in terms of a fact pattern in which the standards for imposing liability were met (as opposed, e.g., to simple confusion about whose job it was to deal with a registration default notice).
N.B. Chapter 718 governs condominiums and Chapter 719 governs cooperative owned real estate, and these generally run more or less parallel to Chapter 720 for home owner's associations discussed above, but there could be subtle differences that I overlooked.