Lets say that somehow, it is proven that Obama is ineligble to be President. Then what?
Does that invalidate the whole election?
Or does Biden become President?
Uncharted ground. I seriously doubt that anyone ever anticipated that it could happen.
My view ... as a non-lawyer, strict constructionist: The law establishes who may be president. If Obama is shown to be (and have been) ineligible, then his election is null and void. The Constitution says, "No person except a natural born Citizen, or a Citizen of the United States, at the time of the Adoption of this Constitution,
shall be eligible to the Office of President; neither shall any Person be eligible to that Office who shall not have attained to the Age of thirty-five Years, and been fourteen Years a Resident within the United States."
Regarding the Veep assuming office, the Constitution says, "In case of the removal of the President from office or of his death or resignation, the Vice President shall become President." (25th Amendment)
My argument is akin to the distinction between a divorce and an annulment. If you get a divorce, you were married and now you're not. If you get an annulment ... you were never married because the marriage is voided. If Obama is not eligible to be President, then his election has to be voided. Everything he signed up until that time is invalid. His appointments are invalid. Biden does not become President, because Biden was elected as VP in partnership with a President whose election was invalid. That makes Biden's election also invalid. Even if Biden's election is NOT voided -- you can't "remove" from office someone who was legally never IN office, so the Constitutional provisions for succession cannot apply.
I think the office would have to be vacated and a special election conducted to fill out the term.