A pre-rooted 184.108.40.206 ROM for the Fire TV 2 was released a couple of weeks ago and now Rbox has released ROMs for the 1st generation Fire TV and the 1st generation Fire TV Stick. These ROMs will allow owners of rooted 1st-gen device to update to the latest software version without losing root access.
You can find the download links to these new ROMs at the top of the Fire TV 1 and Fire TV Stick 1 XDA Forum threads. Please be sure to donate to Rbox to support him and thank him for continuing to keep all of our rooted devices updated.
Use TWRP custom recovery, by following this guide, to install these ROMs on your rooted device. Fire TV Stick 1 owners may need to refer to this guide to be reminded how to get into and control TWRP using a PC.
As always, these ROMs cannot be used to root your device. Your device must already be rooted to install these ROMs. To see if your device’s software version is rootable and how to root it, see my guide here.