Does a replacement effect that says 'if a creature would die, exile it instead' stop death triggers from happening?

A replacement effect that exiles a creature instead of letting it die changes the event itself before it happens. Because the creature moves to exile rather than to the graveyard, it never actually 'dies' by the game's definition. Under CR 700.4, a creature 'dies' only when it is put into a graveyard from the battlefield. If it goes to exile instead, that condition is never met.

Replacement effects are governed by CR 614.1. They intercept an event and replace it with a different one. The original event — moving to the graveyard — never occurs at all, so nothing that triggers 'when [this creature] dies' or 'when a creature dies' gets to trigger (CR 603.2 requires the trigger condition to actually occur).

This is different from a triggered ability that triggers before a creature dies (like a leaves-the-battlefield trigger), which would still function because the creature is still leaving the battlefield — just to exile rather than the graveyard.

Concrete example: Your opponent controls a Sedgemoor Witch and attacks. You block with a 2/2 and cast Leyline Binding, but suppose an effect says 'if Sedgemoor Witch would die, exile it instead.' When the combat damage kills it, it goes to exile. Sedgemoor Witch's own death trigger ('when Sedgemoor Witch dies, create a Pest token') does not trigger, because the Witch never entered the graveyard and therefore never 'died.'

Note: 'Leaves-the-battlefield' triggers (CR 603.6c) still see the creature leave the battlefield, so those would still trigger. Only triggers specifically conditioned on the word 'dies' or 'is put into a graveyard from the battlefield' are blanked by the replacement.