You'd need an impeachment, but there'd obviously be no conviction in the Senate (whether for the documents or some kind of made up incompetency charge.)
And while we're on about the ballot, some highlights from yesterday's Supreme Court hearing:
Justice Thomas: 14th Amendment applying to president would've allowed the South to disqualify presidential candidates from the North.
Justice Kagan: 14th Amendment is designed to take power away from states, not give them more power.
President not likely considered "officer" in context of 14th because:
1. President commissions all "officers". President can't appoint themselves.
2. Something about Congress not being able to hold other offices, yet not excluded from being Speaker or w/e. Not everyone who holds office is an officer. It's a mess.
3. Despite obvious ambiguity, President isn't specifically listed with the others, was listed in earlier drafts.
14th Amendment disqualification is different from other presidential qualifications because it can be waived with 2/3rds vote. States can't block candidate from running by assuming it won't be waived, the same way they can't block a candidate based on current residency (which they can change just before being sworn in,) or an under-aged candidate who will be of age upon taking office.