The Logitech Harmony Smart Hub & Remote has just gone on sale for $99.99 on Amazon. I don’t usually post many non-app deals, but this is the cheapest I’ve ever seen this sell for, and it’s one of the few universal remotes compatible with the Fire TV. Logitech is great about updating their hardware and added support for the Fire TV back in June. The smart hub included with this remote connects via bluetooth to the Fire TV. The remote and free smartphone app communicate with the hub, which then relays the commands to the Fire TV. Since the Fire TV uses bluetooth, and not traditional IR or RF signals, this is one of the only ways to get a universal remote to communicate with the Fire TV.
I gave up on my Harmony One/ Flirc combo and got one of these. Pretty great! The software has a FireTV preset and everything is very responsive.