If a creature dies with a death trigger and a replacement effect that moves it to exile instead, does the death trigger still fire?

A creature 'dies' only when it moves from the battlefield to the graveyard (CR 700.4). Death triggers — abilities that say 'when this creature dies' or 'when a creature dies' — use this specific definition. If the creature never reaches the graveyard, it has not died in the rules sense.

Replacement effects (CR 614.1) modify how an event occurs before it happens. An effect like 'if [this creature] would die, exile it instead' replaces the entire 'move to graveyard' event with 'move to exile.' Because the original event (going to the graveyard) never takes place, there is nothing to trigger a death trigger on (CR 614.5).

It is important to note that state-based actions still try to move the creature — for example, a creature with lethal damage is put into the graveyard as a state-based action (CR 704.5g). The replacement effect intercepts that specific move, redirecting it to exile before it can be completed as a death.

Concrete example: Your opponent controls a 2/2 with the ability 'When this creature dies, draw a card.' You enchant it with a card that says 'If enchanted creature would die, exile it instead.' When the 2/2 receives lethal damage, state-based actions try to move it to the graveyard, but the replacement effect sends it to exile instead. It never dies, so the 'draw a card' trigger never fires.

Contrast this with a trigger that says 'when this creature leaves the battlefield' — that trigger WOULD fire because exile is still leaving the battlefield (CR 700.4 distinguishes 'dies' from the broader 'leaves the battlefield').