Does a replacement effect that changes where a card goes to the graveyard prevent death triggers from firing?

A 'dies' trigger is defined in CR 700.4 as a triggered ability that uses the phrase 'dies,' which means 'is put into a graveyard from the battlefield.' If a replacement effect changes the destination so the card goes somewhere other than the graveyard — for example, exile instead — then the card never 'dies' by the game's definition, and any triggered abilities that watch for that event will not trigger.

Replacement effects that redirect a card (CR 614.1) modify the event as it happens, meaning the original event (going to the graveyard) simply never occurs. Because the trigger condition ('put into a graveyard from the battlefield') is never met, there is nothing to trigger (CR 603.2).

Common sources of such replacement effects include cards like Leyline of the Void or Rest in Peace, which exile cards instead of allowing them to go to the graveyard. If your creature with a 'when this creature dies' ability is affected by Rest in Peace, that creature is exiled rather than placed in the graveyard, so its own death trigger — and any other 'dies' triggers watching for it — will not fire.

Concrete example: You control a Reassembling Skeleton (which has no relevant ability here) and your opponent controls a creature with 'Whenever a creature your opponent controls dies, you gain 1 life.' If Rest in Peace is on the battlefield and your Skeleton would die, it goes to exile instead. Your opponent's life-gain trigger never fires because the creature never went to the graveyard.

Note that this only applies to replacement effects that redirect to a non-graveyard zone. If a replacement effect merely changes something about how the card enters the graveyard (e.g., entering face down), the card still goes to the graveyard and death triggers still fire normally (CR 614.1c).