The 12th Amendment says no one may hold the office of VP who is not eligible to be President. Obama is not eligible as he has already had two terms.
That is an open question. The twelfth amendment says no person shall serve as VP who is constitutionally ineligible to serve as president. However, the 22nd amendment just says no person is eligible to be ELECTED president for more than 2 terms. There is nothing in the 22nd amendment to prevent a 2 term president from being elected Vice President. Since the 22nd refers only to elected to president, not becoming president by succession. So by this, Obama or better yet TRUMP could be elected twice and then run as VP, win the election then become President upon death or resignation of their running mate. My, wouldn’t their heads explode if that were the case with Trump !!!
Thanks for the information/education!