Does a triggered ability still go on the stack if the permanent that triggered it leaves the battlefield before the trigger resolves?

When a triggered ability triggers, it is placed on the stack the next time a player would receive priority — even if the permanent that generated that ability is no longer on the battlefield at that point. The trigger and the permanent are separate objects once the ability is on the stack (CR 603.3).

Triggered abilities are independent of their source once they have triggered. The source's existence is only relevant at the moment the trigger condition is met. After that, the ability exists on its own as a separate stack object (CR 112.7a). This is sometimes called the 'trigger leaves, ability survives' principle.

There is one important caveat: if the ability's effect requires the source to be in a particular zone at resolution (e.g., 'return this card from your graveyard to your hand'), the ability still resolves but that part of the effect may simply fail to do anything if the source isn't there. The ability itself is not countered (CR 603.10).

Concrete example: Your Banisher Priest is destroyed by your opponent in response to its 'when this enters the battlefield' triggered ability still on the stack. That trigger is still on the stack and will still resolve, attempting to exile the target creature — because the ability was already placed on the stack before the Priest died.

In summary, the golden rule is: once a triggered ability is on the stack, it exists independently of its source and will resolve normally unless countered by a specific effect.