Now that rbox has worked out the issues with TWRP for the Fire TV 1, he finally had time to release a pre-rooted ROM of the latest 184.108.40.206 (550145120) software version for both the 1st and 2nd generation Amazon Fire TV. You can download the ROM for the Fire TV 1, labeled 220.127.116.11_r1, here and the ROM for the Fire TV 2, labeled 18.104.22.168_r2, here, and use these instructions to install them through TWRP custom recovery.
These new ROMs will allow Fire TV owners with a rooted device to update to the latest software version without losing root. These ROMs will not help you root a Fire TV, since the Fire TV must already be rooted to install these ROMs. To see if your Fire TV can be rooted in order to install these ROMs, see my Rooting Starters Guide.
Now that TWRP and the latest pre-rooted ROMs for both the 1st and 2nd-gen Fire TV are stable and released, rbox will hopefully continue working on TWRP and ROMs for the Fire TV Stick. If you have the means, please consider donating to rbox to thank him for keeping our rooted Fire TVs updatable, and/or to encourage him to continue his efforts with the Fire TV Stick.