If a candidate who is not qualified to be President because of their age or citizenship at birth, would the runner up be inaugaurated? Would a new election be held? Would their running mate be sworn in?
The 20th Amendment provides that if the President-elect does not qualify on inauguration day, the Vice President-elect acts as President (i.e. serves as "Acting President", but doesn't actually become President) until a President qualifies.
20th Amendment, section 3:
If, at the time fixed for the beginning of the term of the President, the President elect shall have died, the Vice President elect shall become President. If a President shall not have been chosen before the time fixed for the beginning of his term, or if the President elect shall have failed to qualify, then the Vice President elect shall act as President until a President shall have qualified; and the Congress may by law provide for the case wherein neither a President elect nor a Vice President elect shall have qualified, declaring who shall then act as President, or the manner in which one who is to act shall be selected, and such person shall act accordingly until a President or Vice President shall have qualified.
In the case where the President-elect is of insufficient age or residency in the US, that means the Vice President-elect will be Acting President until the President-elect reaches the required age or period of residency in the US, if it happens during the term, at which point the President-elect becomes President.
In the case where the President-elect is not a natural-born citizen, they will never qualify, in which case the Vice President-elect will be Acting President for the entire term.
While this is a hypothetical question with no historical basis or precedent, I will attempt to answer. Realize that reasonable people could disagree with this assessment. Assume that this was a fraud perpetuated on the American electorate and further assume the ineligible President did not resign in disgrace or embarrassment.
I believe that Vice-President and cabinet would invoke the 25th Amendment declaring the sitting President unable to perform the duties of the Presidency due to not meeting the Constitutional requirements of the office. The text reads:
Section 4. Whenever the Vice President and a majority of either the principal officers of the executive departments or of such other body as Congress may by law provide, transmit to the President pro tempore of the Senate and the Speaker of the House of Representatives their written declaration that the President is unable to discharge the powers and duties of his office, the Vice President shall immediately assume the powers and duties of the office as Acting President. Thereafter, when the President transmits to the President pro tempore of the Senate and the Speaker of the House of Representatives his written declaration that no inability exists, he shall resume the powers and duties of his office unless the Vice President and a majority of either the principal officers of the executive department or of such other body as Congress may by law provide, transmit within four days to the President pro tempore of the Senate and the Speaker of the House of Representatives their written declaration that the President is unable to discharge the powers and duties of his office. Thereupon Congress shall decide the issue, assembling within forty-eight hours for that purpose if not in session. If the Congress, within twenty-one days after receipt of the latter written declaration, or, if Congress is not in session, within twenty-one days after Congress is required to assemble, determines by two-thirds vote of both Houses that the President is unable to discharge the powers and duties of his office, the Vice President shall continue to discharge the same as Acting President; otherwise, the President shall resume the powers and duties of his office
There is a scenario where this can happen. If the president and vice president were unable to serve under some circumstance, like both dying in a plane crash. The next person in succession would be the speaker of the house and the house requires a minimum age of 25 and citizenship of the previous 7 years. The house may have a rule requiring the speaker to have the qualifications for the president but this is not a constitutional requirement.
A 25 year old naturalized speaker of the house is not likely but the house was attacked by Puerto Rican separatists during the Truman administration. There is always a possibility of a more serious attack and the only surviving members of both branches, Congress and the Cabinet, are some 25 year old naturalized citizens.