Does a triggered ability that triggers on death still go on the stack if the creature is exiled instead?

A 'dies' triggered ability uses the word 'dies,' which is a defined game term. According to CR 700.4, 'dies' means a creature card is put into a graveyard from the battlefield. If a creature is exiled rather than put into a graveyard, it has not 'died' in the rules sense, so the trigger condition is simply never satisfied.

This matters most when a replacement effect changes where the creature ends up. For example, if a creature has a triggered ability that says 'When this creature dies, draw a card,' but an opponent controls a Rest in Peace (which replaces all cards going to graveyards with exile), the creature goes to exile instead of the graveyard. Because it never touched the graveyard, the 'dies' trigger never fires and you draw no card.

Replacement effects (CR 614) modify events as they happen. The exile replacement effect means the 'put into graveyard' event never occurs at all, so there is no moment for the 'dies' trigger to look at. This is different from a trigger that fires and then the card moves — the card simply goes straight to exile.

The same logic applies to effects like Leyline of the Void or Anafenza, the Foremost's ability: any creature that would go to a graveyard is exiled instead, shutting off all 'dies' triggers on those creatures.

Concrete example: Your Midnight Reaper (which draws a card when a nontoken creature you control dies) attacks into a blocker. Your opponent flashes in Containment Priest, and the Reaper would be put into the graveyard — but Containment Priest exiles creatures that would otherwise enter under certain conditions. If a separate exile replacement effect is in play redirecting Reaper to exile, Midnight Reaper's triggered ability never triggers and you draw nothing.