However, I believe if quo warranto finds Obama ineligible to be president, he will be removed from POTUS by the courts, and Biden would be installed because president-elect Obama was never eligible.
Biden would be elevated, it would seem fairly certain. But I have seen comments saying that whether Obama was “never” eligible could depend on whether he was ineligible due to intentional fraud (hiding foreign birth), in which case his ineligibility would be retroactive to inauguration, or ineligible due to a “case of first impression” SCOTUS ruling that he was not NBC due to having a foreign citizen parent. The latter type of ineligibility might not be retroactive. Worst case it might be applied only prospectively, giving Obama a Mulligan! (Heaven fore-fend!)