But the 20th amendment seems to say that the House would continue to attempt to qualify a President, when it says:
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;
But of course they don't have many choices, only among the top 3 (eligible??) electoral vote getters. How long can it take?
Of course if the declaration of ineligibility happens after the electoral votes are opened and counted, the Constitution is not really clear what should happen.
Of course the founders likely never thought someone not "eligible to the office" would even run, (or the equivalent at the time, in which the candidate didn't actually run in the modern sense) let alone be selected.
One thing is for sure:
All Hell Would Break Loose
That is something the Supreme Court will want to avoid, not just the riots that would ensue no matter when the Obamanation is dethroned.
The election never goes to the House if there is a majority winner in the EC. The 12th Amendment spells out the conditions for the House to choose the President but makes no reference to a qualification process. The 20th Amendment suggests that the qualification of a President takes place after the EC votes are counted ("President elect shall have failed to qualify") whether or not the House becomes involved. I find no other Constitutional reference to the qualification process. And there isn't an eligibility requirement to run for President, only to be President. What we seem to have here is a President-presumptive (Obama) that has a majority of the EC votes who will become the President-elect after the EC votes are counted, but who might fail to qualify due to his place of birth and consequently never actually attain the office. If you consider the case where a candidate would not have achieved the required age of 35 until Feb 1, 2009, or met the residency requirements until then, then the VP elect become the VP but would act as President for those 11 days.
However, there is another possibility: what if the election goes to the House and they refuse to choose McCain, their only possible choice? Basically, they sit on their hands and do nothing. Then, clearly by the 12th Amendment Biden is Acting President until they do choose or there is another election in 2012.