If not eligible, Obama and co-conspirators/abettors must be punished. Legislation signed by the One will stand as his signature is not necessary; if he does not sign, the legislation becomes law any way. Executive ordeers, appointments and regulations made by the appointees, fines levied as a result of those regulations etc. is another thing. It would be a Constitutional cricis and schmozzle of Biblical proportions.
Actually, IMHO, the legislation signed by the Obama would not stand, because the US Constitution says that every bill must be presented to the president before it can ever become law. If Obama is ineligible for the presidency, then he was never legally president, so if Congress presented legislation to Obama, it would not satisfy the US Constitution's requirement in Article I Sec. 7 that all legislation be presented to the president before it becomes law. Thus any unsigned legislation that would be sent to Obama would not automatically become law without his Obama's signature, because the legislation was never presented to the president as required by the US Constitution.
Unsigned/unvetoed legislation only becomes law without the president's signature, if ten days (excluding Sundays) elapse after it was presented to the president. If Obama is ineligible for the presidency, unsigned legislation would not automatically become law, because it was never presented to the president as required by the Constitution.
US Constitution Article I. Sec.7
shall, before it become a Law, be presented to the President of the United States;...If any Bill shall not be returned by the President within ten Days (Sundays excepted) after it shall have been presented to him, the Same shall be a Law, in like Manner as if he had signed it, unless the Congress by their Adjournment prevent its Return, in which Case it shall not be a Law.