Answer:
Actually Pan was never in the film at all, there was a faun but his name is not Pan. Pan was a Greek god in the form of a faun and this can explain why the faun is a faun and how he is the guardian to the Labyrinth to look for the princesses soul. Back to the question, no Pan doesn't die and nether does the faun with many names.
Pan (as in the character) is a roman mythological creature that is half human and half goat (faun), that is why he is dressed with horns etc. In roman mythology he lives in the forest and is very rarely seen, he also plays the Panpipes. Also in Greek Mythology the word "pan" is the god of Shepard and hilltop animals (goats) so the two interlock.