Can I play Gonti, Lord of Luxury's exiled card after he leaves the battlefield?
As said on Gonti, Lord of Luxury 's gatherer page :


  You may look at and cast that card (and spend mana as though it were mana of any type to do so) even if Gonti leaves the battlefield. If another player gains control of Gonti, that player can’t look at or cast the card, and you still can.


If it comes back to the battlefield later, it'll be a completely different object, and won't affect the first card exiled, you'll still be able to cast it.