I suspect if Obama was found ineligable after he takes office the normal chain of sucsession would apply. VP Biden, then Speaker Pelosi.
If Obama was found ineligable prior to the Electorial College vote, I think the Obama electors would fell free to vote for whoever they wanted. They would either get together and as a group decide to all vote for an alternative (my guess it would be Biden, but who really knows at that point). If they couldn’t and nobody got the majority it would go to the senate (or is it the house?)
The gotcha comes if Obama is invalidated in the no-man’s land between when the electors have voted (for Obama), but before he has been sworn in. Does it go by the chain of sucession or does it go to congress?
It goes by the Constitution, which has addressed this in the Amendments. If a President-elect fails to qualify, the Vice President-elect holds the office until a President shall have qualified.