How do I customize the alert sound for an iPhone timer?
If you're able to save the file you'd like to play as a ringtone, then the iPhone can play it once the timer hits zero. The big limitation here is that it'd need to be ~35 seconds in length, but depending on your specific need, that might be enough.

There are a few apps on the App Store that will allow you to create ringtones from music files, or you could do it yourself in Garageband. The linked instructions are for Garageband '11, but searching "garageband ringtone" at Apple support will yield links for various other versions of Garageband - the process is pretty much the same.