Could an opponent cast a card with the same name as a card exiled by Circu, Dimir Lobotomist after you defeat the owner of the card in a Free for All?
Since there is some confusion about the original wording of the question, let's first assume you control Circu, Dimir Lobotomist, used it to exile a couple of spells from an opponent's library and that opponent is now defeated.
800.4a When a player leaves the game, all objects (see rule 109) owned by that player leave the game ...
The exiled cards owned by the defeated player leave the game, and don't count for Circu's ability anymore:
607.2a If an object has an activated or triggered ability printed on it that instructs a player to exile one or more cards and an ability printed on it that refers either to "the exiled cards" or to cards "exiled with [this object]," these abilities are linked. The second ability refers only to cards in the exile zone that were put there as a result of an instruction to exile them in the first ability.
Your other opponents can cast those spells again (unless you've also exiled said card from a library of an opponent who is still in the game).
If you meant to ask what happens when an opponent controlling Circu, Dimir Lobotomist is defeated; the exiled cards stay exiled, but Circu's static ability stops functioning: it's no longer on the battlefield, and you (and other opponents) can cast spells with the same name as the exiled cards once again.