Earlier threads that have addressed (and pretty much exhausted) this topic:
American President and terms in office
Could Kerry choose Clinton as a running mate
Could Clinton become VP?
I can see three possible views about whether a twice-elected president can become vice-president:
A twice-elected president is ineligible for the vice-presidency under any circumstances. The 22nd amendment makes him or her ineligible for election as president, so the 12th amendment makes him or her ineligible for election as vice-president. The 25th amendment, the only other route to the vice-presidency, is silent about eligibility but implicitly imports the eligibility requirements from the 12th and 22nd amendments. (To put it another way, the 22nd amendment means that being re-elected as president exhausts one’s eligibility, not only for purposes of election but for all purposes.)
A twice-elected president is ineligible for the vice-presidency by election, but not necessarily by way of the 25th amendment. The 12th amendment’s eligibility clause imports the 22nd amendment’s limit on electability, but the 25th amendment does not involve an election, so a twice-elected former president can become vice-president if the incumbent president nominates and both houses of Congress confirm him or her.
A twice-elected president is eligible for the vice-presidency under any circumstances. The 22nd amendment’s literal terms apply only to election as president, not to election as vice-president or to succession to the presidency.
Views 3 and 2 are more literal (that is, more textually defensible) than View 1, but Views 1 and 2 make more sense than View 3 in light of the 22nd amendment. The amendment says only that “[n]o person shall be elected,” but surely its intent was to keep a demagogue (or any individual, for that matter) from monopolizing the executive branch indefinitely. If a twice-elected president is eligible for election as vice-president, and is then eligible for succession to the presidency, what would stop a popular term-limited president from running for vice-president on a ticket with a figurehead who will take office then immediately step aside? The constitutional gymnastics are a little more complicated, but the same policy question can arise in the case of a twice-elected president chosen as vice-president under the twenty-fifth amendment.
A statutory succession to the presidency under the Presidential Succession Act raises substantially the same issues, since the Act’s provisions “apply only to such officers as are eligible to the office of President under the Constitution.” 3 U.S.C. § 19(e). (Interestingly, there was a book written more than 30 years ago – Line of Succession by Brian Garfield – that deals with some of these issues. The vice-president and speaker are killed in a terrorist attack, the president-elect is kidnapped then killed, and the president pro tem is politically unacceptable to both political parties, so the defeated outgoing president schemes to amend the Succession Act so that he can stay in office. The book sidesteps the 22nd-amendment issue, though, because the outgoing president has served only one term.)
Short of a constitutional amendment or a court case, I suppose that there can be no definitive answer until somebody tries it.