Biden may or may not become the next president of the US if Obama is declared uneligible for the presidency. But because Biden ran on the same ticket as Obama, he may not be vice-president if Obama could not be the president.
Its one thing if the president is disabled (car wreck, coma, etc) but another if he committed fraud to win an election. Fraud about his eligibility. The constitution does plan for a disabled president but not a fraudulent one.
The 20th amendment does make a provision for times when the president can not hold office after the election -- because of death or disability, not because the president committed fraud to be eligible for president.
I would also note that President Obama and Vice-President Biden ran as a single ticket -- Biden became vice president because he ran on the same ticket as Obama.
This may indicate that Romney is either the vice-president or president (see the 12th amendment), which is perfectly legal. As fair as I can tell, the practice of having a president and vice-president of the same party is not from a constitutional requirement, just the way that both parties have done things for many years.
Also, according to the 12th amendment "no person constitutionally ineligible to the office of President shall be eligible to that of Vice-President of the United States," and the question begs, how does this apply to ticket candidates? Do both parties need to be eligible in order to have a valid ticket? Or does only one party need to be eligible in order to have a valid ticket?
I would think that both parties need to be eligible in order to have a valid ticket, but the courts will decide (if it comes up).