Would Obama be allowed to serve as VP? If so, I can see that happening. It would take the 35% support Trump has at present back to the 8% level with the 2016 election, and again make it a marginal election, capable of going either way.
The pertinent issues at hand are:
—No President can serve more than ten years or be elected more than twice. (22nd amendment)
—No person ineligible to be President can be Vice President.
In theory, Obama (as with Bill Clinton and George W. Bush) still have two years of eligiblity, but cannot be elected President. What is certain is that, if the originally elected Vice President vacates the seat, any of those term-limited Presidents could be
appointed to the vacancy by consent of the Senate. If, then, the President vacates the seat, the term-limited President would be limited to two more years in office, when he would then himself have to vacate the seat in favor of someone else.
Also consider: if someone has been elected Vice President, it is not counted against that "elected more than twice" count (see: any former VP who has run for President and won, such as George H. W. Bush). So, it appears, that Obama could run as a VP, even after using up his two Presidential elections, then succeed to the seat for two years if Biden steps aside.
Now here's where things get tricky: the Presidency is a four-year term. If the Vice President is only eligible for two years from the start, can he legally be eligible for the seat knowing full well he would not be eligible to fulfill the role of President for the full term if he succeeds?
And if he does run again, who would have standing to challenge it? As we saw with the birther fiasco, almost every court case regarding Presidential eligibility has been thrown and laughed out of the courts.