First of all, most of the discussion has been silent on the simple fact that Obama's mother is clearly a natural born citizen herself. I thought children of US citizens were automatically natural born regardless of where they were born. Apparently there is some minority legal scholarship that says otherwise, but you'd think a child of an American mother would be pretty clearly a US citizen. US Embassies around the world will get a baby of such US citizens a passport. So wherever Obama was born (pretty clearly Hawaii) and whatever other legal contortions can be made, presumably if this even gets to the full Supreme Court, Obama's position is safe.
But assume for a minute it weren't? What if, after all, Obama were not constitutionally qualified to be president? What then? Would the rules of succession apply? I suspect if Obama were ruled ineligble before the Electoral College meets, then the Electors would decide on the next president. If the ruling comes afterwards, what then? Probably the rules of succession. Either way, what a mess!
That's why it won't come to that. It would be even more Mickey Mouse than Thailand shutting down it's main Bangkok airports at the height of the travel season!