If a permanent returned to the battlefield with unearth would leave the battlefield for any reason, it's exiled instead—unless the spell or ability that's causing the permanent to leave the battlefield is actually trying to exile it! In that case, it succeeds at exiling it. If that spell or ability later returns the card to the battlefield (as Static Prison might, for example), the permanent card will return to the battlefield as a new object with no relation to its previous existence. The unearth effects will no longer apply to it.