The OP questioned what would happen if, after a presidency concludes, the President admits to not being a citizen, thus “ineligible.” Some of the comments have broadened the scope of the question to what would happen if, during his/her presidency, the President is found to be “ineligible.”
To keep this in GQ, this post addresses a particular issue of Constitutional interpreation… what it means to be “eligible to the Office of President” as recited in Article II, Section 1.
First, it’s kind of a strange wording “eligible to the Office” rather than “eligible to be President” or, as stated in Article I regarding the House and Senate respectively, “No Person shall be a Representative” and “No person **shall be **a Senator.” The word “eligible” is only used in the context of the presidency. The Twelvth Amendment further adds that “no person constitutionally ineligle to the office of the President shall be eligible that of the Vice-President.” Does it recite eligibility to be President, to be elected President, to be *sworn in *as President, or something else?
Checking Webster’s Dictionary of 1828, we note that “eligible” has three definitions. First, “Fit to be chosen; worthy of choice, preferable.” Second, “Suitable; proper; desirable.” And most relevantly, “Legally qualified to be chosen; as, a man is or is not eligible to an office.” So, we read “eligible” to mean, legally qualified to be chosen and figure that once you’ve been “chosen,” you don’t have to meet the qualifications anymore. This really only applies to the President who can, I guess… denounce citizenship or expatriate while in office and remain President? Once you’re a certain age, you can’t get go backwards. This supports the “be elected” interpreation.
With respect to the Senate, there have been a few constitutionally ineligble Senators. See U.S. Senate: Youngest Senator. In one case, the Senate determined that the age requirement for eligibility applied at the time of oath-taking, rather than the time of election or the time the term begins. This supports the “be sworn in” interpretation, however, alternative language is used (see supra) with respect to the Legislative and Executive branches.
Either way, the President needs only establish eligibility to *become *president, not to *be *President. The question then remains, can a President be removed because he/she was not a natural born citizen at the time of “choosing” (electoral college vote, I’d guess)? Probably not.
Article II, Section 4 states that the President “shall be removed from Office on Impeachment for, and Conviction of, Treason, Bribery, or other high Crimes and Misdemeanors.” Note that “constitional ineligibility at a prior date” is not one of those listed. Thus, once you’re President, you’re President… previously eligible or not. As a caveat, someone could argue that you have to claim on some government form that you’re eligible to be President by filing the appropriate papers to run (I haven’t looked this up), or that you’ve constructively stated so, which could be construed as perjury, which might be a “high crime.”
Still, to answer the original posters question… nothing.