In Philadelphia, the hats are collected and displayed.
There is a hat trick display in the First Union Center concourse. The hats are piled up in a section of wall which is covered in glass, with separate piles of hats and plaques noting the accomplishments.
Hats that aren’t on display are stored in big garbage bags. The PR folks in Philly count them and pass the word around about “record hat tosses” after so-and-so’s last hat trick.
Many arenas collect them and trash them.