Does a triggered ability that triggers on a creature dying still trigger if the creature is exiled instead?

The word "dies" is a defined game term in Magic. According to CR 700.4, "dies" means a creature card is put into a graveyard from the battlefield. This is a precise, narrow definition — it does not include exile, the command zone, or any other destination.

If a creature would normally die but is instead exiled — for example, due to a replacement effect like the one from Leyline of the Void or Rest in Peace — it never touches the graveyard. Because the creature did not move to the graveyard, it did not "die" by the rules definition, and any ability that triggers "when [creature] dies" will not trigger at all (CR 603.2, 700.4).

Replacement effects are the most common cause of this. Under CR 614.1, a replacement effect modifies what happens to the creature, substituting exile for the graveyard. Since the game event of "dying" never occurred, there is nothing for the triggered ability to respond to.

Concrete example: You control a Zulaport Cutthroat ("Whenever another creature you control dies, each opponent loses 1 life..."). Your opponent casts Swords to Plowshares targeting your Grizzly Bears. Because Swords to Plowshares exiles the creature rather than destroying it, the Bears are exiled — they never enter the graveyard, so Zulaport Cutthroat's triggered ability does NOT trigger.

Note: if a card uses broader language like "when [creature] leaves the battlefield" or "when [creature] is put into a graveyard or exiled," it may cover exile explicitly. Always read the exact wording of the triggered ability.