Does a triggered ability still resolve if the source leaves the battlefield?

A triggered ability is placed on the stack as its own object the moment its trigger condition is met (CR 603.2). Once it is on the stack, it is completely separate from the permanent or object that generated it. The source leaving the battlefield does not remove the ability from the stack.

This is governed by CR 112.7a, which states that abilities on the stack are not affected by what happens to their source after the ability triggers. The ability will continue to exist on the stack and will resolve normally when it reaches the top and both players pass priority.

There is one important exception: if the ability's effect requires information about the source (such as its power or toughness) at the time of resolution, and that information is no longer available, the game uses the last known information about that object (CR 112.7a, 608.2b). This can affect how much damage is dealt, for example, but does not stop resolution.

Example: You control Darksteel Colossus and it has a triggered ability that says 'When this creature dies, draw a card.' If an opponent responds to that trigger by exiling the Colossus from the graveyard, the triggered ability is already on the stack and will still resolve — you still draw a card.

So the key takeaway is: triggers 'go off' and live on the stack independently. Removing the source after the trigger is on the stack is generally too late to stop it from resolving.