“some believe Frank Marshall Davis may be his biological father.” If so, then he is most certainly qualified to be President. That would mean both his parents are American and some sort of US/Hawaiian birth certificate was requested shortly after his birth. I think that’s the end of the story, if true. I doubt that is the case, however.
The media would spin mightily to make Obama into the victim of right-wing persecution (see the “Clinton-Lewinsky” saga).
But I think that some independents would ask themselves when he knew of this - was it before he published his book about his father? Was it before he made all those speeches about his parents?
I don't want to spin this out too much (I guess I did already), but it would still be a political problem.