We have learned that many important constraints on elected officials are norms rather than rules. In an effort to turn some norms effectively into rules, can states add to their requirements for a person to be on the ballot? For example, 5 years of tax returns must be released to the public?
In US Term Limits v. Thornton, 514 US 779, the Supreme Court recites the constitutional requirements for holding national office as representative or senator, which pertain to age and citizenship+residence. The court held that an Arkansas law "prohibiting the name of an otherwise-eligible candidate for Congress from appearing on the general election ballot if that candidate has already served three terms in the House of Representatives or two terms in the Senate" was unconstitutional, and "Allowing individual States to adopt their own qualifications for congressional service would be inconsistent with the Framers' vision of a uniform National Legislature representing the people of the United States". The same reasoning would hold for setting qualifications for the office of President. However, in Bush v. Gore, 531 U.S. 98, the court pointed out that "The individual citizen has no federal constitutional right to vote for electors for the President of the United States unless and until the state legislature chooses a statewide election as the means to implement its power to appoint members of the electoral college", noting that "The State, of course, after granting the franchise in the special context of Article II, can take back the power to appoint electors".
It is uncontroversial that states have the right to impose conditions on ballot access, and that a person does not have an automatic right to be on the ballot simply by applying for candidacy. You might argue that being required to 20,000 signatures on a petition constitutes imposing an unconstitutional qualification on a candidate, but the Supreme Court has so far not clearly found such ballot access provisions to be unconstitutional. Still, there is some room for argumentation. This page lists the 11 decisions regarding ballot access. In the first case, Williams v. Rhodes, 393 U.S. 23, the court overturned a requirement to garner 15% of the total votes in the last gubernatorial election for a new party to be on the ballot, finding that the laws "are invidiously discriminatory and violate the Equal Protection Clause", but later decisions tended to accept requirements with later filing dates or lower percentages. Filing fees are generally seen as an unconstitutional burden (Bullock v. Carter, 405 U.S. 134, Lubin v. Panish, 415 U.S. 709). It is hard to see how requiring a candidate to submit their tax returns could be seen as an unconstitutional burden.
When it comes to state elections, there is even less of an argument that a state cannot impose such requirements (on dog-catcher, legislator, or governor). Though in an individual state, there might be a constitutional provision speaking against such a law.
@user6726's response is good, but I'd like to point out that the constitutional burden would be unduly put on the 4th Amendment, which gives an implied right to privacy. As it currently stands, the only way the President's tax returns could be revealed to the public is if the President chose to do so or was ordered to due so by due process of the law (issued a warrant by a judge with probable cause standards).
As is the case, the matter has already been settled about traditions of the office of president being legal burdens. Prior to FDR's four terms as President, all Presidents observed two term limit (of the three presidents who would run for a third term, only FDR was voted one) in honor of George Washington refusing a very winnable third term because he did not want to look like a king. The U.S. had to amend the constitution to turn this tradition into law. Since the modern tradition of releasing tax returns is tradition, and not found in the constitution, without an amendment, it is not a legal requirement for elected Federal Office.
Apparently we will soon find the answer tp the question because, according to CNN
Lawmakers in 18 states across the country, including New York, Illinois and Washington, have introduced bills that would require all presidential candidates and their running mates to release their individual tax returns in order to qualify for the presidential primary ballot, according to the National Conference of State Legislatures.