If it did come out that Obama was born in Kenya (which I doubt), Congress would take one look at Joe Biden and think about the possibility of race riots on a much worse scale than after the King assassination, and quickly pass a law redefining "natural born citizen" so that Obama would be covered. If somehow that was impossible, Congress could quickly vote for, and the states ratify, a Constitutional amendment removing the "natural born citizen" requirement. Even if Obama's popularity continues to plummet, the politicians in Congress and in the state legislatures would be terrified at the thought of a race war if Obama was removed from office.
Even if you could get 38 state legislatures to ratify it would take years. Many legislatures don't even meet each year, and others would vote no.