If the Electoral College elects Obama as President and Biden as VP and Congress counts the votes and accepts them, and later (but before the Inauguration) the courts declare Obama not to be a natural-born citizen (and hence not to meet the constitutional qualifications), then there would be a vacancy in the office of the president and the new VP (Biden) would succeed to the presidency immediately upon being sworn in as VP; since Obama would be unable to qualify for the presidency at a later date (given that the qualification he would lack would not be one that could be cured with time, such as would have been the case had he been, say, just under 35 years old or a resident of the U.S. for a bit less than 14 years), Biden would serve as president for the entire term.
If the Electoral College elects Obama president and Biden VP and Congress refuses to count votes for Obama because it deems him not to meet the constitutional qualifications, then no candidate for president would have received 270 EVs and the House would elect the president from among the candidates with the three highest number of EVs, with each state delegation getting one vote. If McCain is the only constitutionally qualified candidate to receive EVs, then the House can either elect McCain or elect no one, and since Democrats control a majority of House state delegations they may elect no one so that Biden serves as president for the entire term (I’m assuming that if less than a majority of members of a state delegation votes for McCain it won’t be deemed as a vote for McCain by such state delegation, and so the Democrats could keep McCain under 26 votes). However, if just one of the Democrats in the Electoral College votes for Hillary (or some other Democrat for president), then she would be eligible for election by the House (assuming that she finishes among the top three candidates). Of course, the GOP could counteract this by having its electors cast a bit over 1/3 of its votes for McCain and a bit less than 1/3 for each of, say, Sarah Palin and Dick Cheney, so the top-3 legal candidates would be Republicans even if a few Democrat electors cast votes for Hillary, but that would take quite a bit of foresight and planning and I don;t think the GOP thinks that the scenario I described above could possibly occur.
If Obama is deemed by the courts to be constitutionally unqualified prior to the Electoral College meeting, then Obama and the Democrats would likely hold a conference call with the 365 Democrat presidential electors and instruct them to vote for Hillary or whomever it is that would be their preferred replacement, and such person would be elected president when the Electoral College meets. Either that or Obama and the Democrats will choose to ignore the Supreme Court and insist on Obama’s eligibility, and the Electoral College will elect him, the Democrat Congress will count the votes for Obama, some notary will sewr Obama in as president and we’ll have a bona fide constitutional crisis in our hands.
One scenario I didn’t mention above was that the courts declare Obama not to be a natural-born citizen after he has already taken office. That would also result in a constitutional crisis even if Obama agreed to resign from the presidency, since the legal status of his appointments and the bills that he purportedly signed into law would be questionable at best.
Interesting thought, if we had the elections over, Hillary only SUSPENDED her campaign when Obama got the nominee. Does this mean anything? Just thinking she might try running again.