In the book and movie Finnic dies while on a mission to kill Snow. In the capitol they had set up traps called pods to keep them out, they can be anything from explosives to deadly animals designed to kill. Finnic dies from one of these pods from being decapitated. Finnic was decapitated from mutts that were mutations of lizard and a form of skinless wolf. If you watch mockingjay part 2 it shows how Finnic die