[Tarkus_]

Best to set up a screen saver in Kodi to stop the Fire TV's screen saver from kicking in.  Since there's no clean way back into Kodi (to exit properly, for instance) once that happens.

Does that require an addon? I did try enabling the built-in options to put the display to sleep as well as shutdown when idle, but they didn't seem to make a difference.

There are screen savers available in Kodi's official repo.  Go to Settings > Interface > Screensaver to download and/or enable a screen saver.