According to this ruling (which, on its face, appears to be legally sound), there are 2 ways:
1) File a challenge to a candidate's qualifications before the candidate is elected; or
2) Go through the impeachment process.
The reality is, at this point, Barack Obama has been elected and sworn in as President of the United States. According to the Constitution, the President of the United States may only be removed from office through the impeachment process. It doesn't matter what high crime or misdemeanor, or treasonous act, the President commits, impeachment is the only Constitutional way to get a President out of office.
That was done, and rejected on "standing" and "ripeness". I understand the ripeness part, there is no injury for someone just being nominated by a party, eligible or not. Perhaps between the election and inauguration would be the "proper" time, although that was attempted as well.