The answer is not nearly as simple and straighforward as the employee v. independent contractor distinction suggested in the question.
People who work for free fit in a number of categories, some lawful and some not, even though the definition of employer under the Fair Labor Standards Act (FLSA) (regulations here) and parallel state legislation is facially very broad (people who suffer to let others work for them, more or less) and many of the exceptions are quite narrow.
The nature of the categories is a nice illustration of how much social context and nuance is buried in a seemingly straightforward black and white legal rule like the minimum wage. Any time there are common practices that a statute doesn't expressly contemplated that aren't routinely the subject to litigation, you should doubt whether the statute really applies to them (although often enough it does), and courts are more likely in these circumstances than most to devise a definitional oriented, or implied, or common law exception to encompass the common practices as being lawful.
The facts in the original question aren't rich enough in character to really determine which category applies, and those facts "smell fishy" as far as whether it would really fit in any legal category of unpaid work, in part, because the motives of the "volunteer" are very unclear.
I'm sure that there are other possibilities that I have not considered and listed, but generally, the legality of working for free is evaluated on a category by category basis. Situations where payment is predominantly in kind (e.g. room and board and clothing) can be complex to analyze and these arrangements are frequently subject to special rules.
The primary statutory minimum wage exemptions are as set forth in the exemptions provision of the FLSA:
§213. Exemptions
(a) Minimum wage and maximum hour requirements The provisions of
sections 206 (except subsection (d) in the case of paragraph (1) of
this subsection) and 207 of this title shall not apply with respect
to—
(1) any employee employed in a bona fide executive, administrative, or
professional capacity (including any employee employed in the capacity
of academic administrative personnel or teacher in elementary or
secondary schools), or in the capacity of outside salesman (as such
terms are defined and delimited from time to time by regulations of
the Secretary, subject to the provisions of subchapter II of chapter 5
of title 5, except that an employee of a retail or service
establishment shall not be excluded from the definition of employee
employed in a bona fide executive or administrative capacity because
of the number of hours in his workweek which he devotes to activities
not directly or closely related to the performance of executive or
administrative activities, if less than 40 per centum of his hours
worked in the workweek are devoted to such activities); or
(2) Repealed. Pub. L. 101–157, §3(c)(1), Nov. 17, 1989, 103 Stat. 939.
(3) any employee employed by an establishment which is an amusement or
recreational establishment, organized camp, or religious or non-profit
educational conference center, if (A) it does not operate for more
than seven months in any calendar year, or (B) during the preceding
calendar year, its average receipts for any six months of such year
were not more than 331/3 per centum of its average receipts for the
other six months of such year, except that the exemption from sections
206 and 207 of this title provided by this paragraph does not apply
with respect to any employee of a private entity engaged in providing
services or facilities (other than, in the case of the exemption from
section 206 of this title, a private entity engaged in providing
services and facilities directly related to skiing) in a national park
or a national forest, or on land in the National Wildlife Refuge
System, under a contract with the Secretary of the Interior or the
Secretary of Agriculture; or
(4) Repealed. Pub. L. 101–157, §3(c)(1), Nov. 17, 1989, 103 Stat. 939.
(5) any employee employed in the catching, taking, propagating,
harvesting, cultivating, or farming of any kind of fish, shellfish,
crustacea, sponges, seaweeds, or other aquatic forms of animal and
vegetable life, or in the first processing, canning or packing such
marine products at sea as an incident to, or in conjunction with, such
fishing operations, including the going to and returning from work and
loading and unloading when performed by any such employee; or
(6) any employee employed in agriculture (A) if such employee is
employed by an employer who did not, during any calendar quarter
during the preceding calendar year, use more than five hundred
man-days of agricultural labor, (B) if such employee is the parent,
spouse, child, or other member of his employer's immediate family, (C)
if such employee (i) is employed as a hand harvest laborer and is paid
on a piece rate basis in an operation which has been, and is
customarily and generally recognized as having been, paid on a piece
rate basis in the region of employment, (ii) commutes daily from his
permanent residence to the farm on which he is so employed, and (iii)
has been employed in agriculture less than thirteen weeks during the
preceding calendar year, (D) if such employee (other than an employee
described in clause (C) of this subsection) (i) is sixteen years of
age or under and is employed as a hand harvest laborer, is paid on a
piece rate basis in an operation which has been, and is customarily
and generally recognized as having been, paid on a piece rate basis in
the region of employment, (ii) is employed on the same farm as his
parent or person standing in the place of his parent, and (iii) is
paid at the same piece rate as employees over age sixteen are paid on
the same farm, or (E) if such employee is principally engaged in the
range production of livestock; or
(7) any employee to the extent that such employee is exempted by
regulations, order, or certificate of the Secretary issued under
section 214 of this title; or
(8) any employee employed in connection with the publication of any
weekly, semiweekly, or daily newspaper with a circulation of less than
four thousand the major part of which circulation is within the county
where published or counties contiguous thereto; or
(9) Repealed. Pub. L. 93–259, §23(a)(1), Apr. 8, 1974, 88 Stat. 69.
(10) any switchboard operator employed by an independently owned
public telephone company which has not more than seven hundred and
fifty stations; or
(11) Repealed. Pub. L. 93–259, §10(a), Apr. 8, 1974, 88 Stat. 63.
(12) any employee employed as a seaman on a vessel other than an
American vessel; or
(13), (14) Repealed. Pub. L. 93–259, §§9(b)(1), 23(b)(1), Apr. 8,
1974, 88 Stat. 63, 69.
(15) any employee employed on a casual basis in domestic service
employment to provide babysitting services or any employee employed in
domestic service employment to provide companionship services for
individuals who (because of age or infirmity) are unable to care for
themselves (as such terms are defined and delimited by regulations of
the Secretary); or
(16) a criminal investigator who is paid availability pay under
section 5545a of title 5;
(17) any employee who is a computer systems analyst, computer
programmer, software engineer, or other similarly skilled worker,
whose primary duty is—
(A) the application of systems analysis techniques and procedures,
including consulting with users, to determine hardware, software, or
system functional specifications;
(B) the design, development, documentation, analysis, creation,
testing, or modification of computer systems or programs, including
prototypes, based on and related to user or system design
specifications;
(C) the design, documentation, testing, creation, or modification of
computer programs related to machine operating systems; or
(D) a combination of duties described in subparagraphs (A), (B), and
(C) the performance of which requires the same level of skills, and
who, in the case of an employee who is compensated on an hourly basis,
is compensated at a rate of not less than $27.63 an hour; or
(18) any employee who is a border patrol agent, as defined in section
5550(a) of title 5; or
(19) any employee employed to play baseball who is compensated
pursuant to a contract that provides for a weekly salary for services
performed during the league's championship season (but not spring
training or the off season) at a rate that is not less than a weekly
salary equal to the minimum wage under section 206(a) of this title
for a workweek of 40 hours, irrespective of the number of hours the
employee devotes to baseball related activities.