No, Biden will remain Obama's VP during Obama's second term, if Obama is reelected. Though I doubt he will run for president in 2016.
I think he would try to run in the primaries, but there will be a lot more "popular" candidates like Warner, Hillary, even Bayh.
I'm sure Biden has his fans, but he's such a lose cannon that he would really have to change his personality in order to really win over his detractors. Biden, like Cheney, is perfect as a tough guy, experienced washington insider. But those qualities aren't going to win you popularity contests across the midwest.
Plus, he would be fairly old, and someone more youthful like Warner or Bayh could energize the crowd moreso than Biden.
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