What happens if Obama wins the election and after he is sworn in, it is learned that he was born on foreign soil and therefore ineligible for election? I think that there would be a Constitutional crisis which would make the 2000 Florida situation look like child’s play. Since Obama and his VP ran as a ticket and not individually, I would guess that the VP couldn’t be sworn in as President because his/her election was not legitimate, either. The scariest part of this scenario would be that the highest undisputed elected official (according the succession order) would the be the Speaker of the House.
If you have just eaten, DO NOT READ THIS:
President Pelosi.
Article II, Section 1:
In Case of the Removal of the President from Office, or of his Death, Resignation, or Inability to discharge the Powers and Duties of the said Office, the same shall devolve on the Vice President,...
A President who is not eligible for the office cannot discharge its Powers and Duties, so the Vice-President would be sworn in.
However, since Barack Obama is a natural born citizen, the situation you imagined cannot happen. See my post #70 above.