What happens if Obama's BC had been a fake? Biden becomes President?
I would assume so. The election was won for the Dem ticket, not just Obama. That's actually one of the reasons we use the electoral college system. If someone were to die at any point between election day and the inauguration, the party can theoretically select a replacement for either, or even both slots. This would theoretically apply in the case of disqualification as well.
Since Obama's already been inaugurated, then it's even less complicated (although there are other issues). Presumably, he'd either step down, or be impeached, and Biden would become president. If for some reason he didn't step down, and wasn't impeached, we'd have a constitutional crisis (because those are the only means for removing a president). Theoretically, the Supreme Court would get involved, but I'm not aware that they have any power to remove him. They could rule that anything he does is in violation of the constitution I suppose, but that starts to get really messy to even contemplate.