I found an Amazon Fire TV game controller at Goodwill for a steal, but I don't have Fire TV. Can I hook this up to a PC, Xbox 360, or Xbox One?
This is partial speculation as I personally don't own one, but here goes:
pc: According to the amazon page q&a, the controller can link to a pc via bluetooth. I assume it will setup as a standard joystick/gamepad.
xbox: Here's where it gets complicated:
If the controller uses Infrared to transmit standard tv codes, I believe that some xbox consoles support minor control (play/pause in movies, etc...)
Only controllers that have been designed to be xbox compatible will work for playing games; typically requiring special boards/chips from microsoft (see off brand controllers such as afterglow) and need to be designed for only that purpose. There may be some other way of playing (like using the xbox windows app) but a direct connection will not work.