The Apple TV app, which provides access to Apple TV+ and iTunes movies, has been updated to support Dolby Atmos audio on newer Fire TV devices.
Step by step
The Apple TV app launched on Amazon's Fire TV platform in late 2019 with several limitations. Support for Dolby Vision HDR was added some time after.
The latest app update also unlocks Dolby Atmos audio, as highlighted by a user on reddit (link). We have it working on Fire TV Stick 4K Max and it should also work on Fire TV Cube and possibly older Fire TV Stick devices that support Dolby Atmos (please check and let us know if you have one).

Apple TV app on Amazon Fire TV
Dolby Atmos immersive audio is available on most Apple TV+ movies and shows including Finch, Foundation and For All Mankind. Many movies on iTunes (547to be exact) are also available with Dolby Atmos.
Also read: Full list: 4K HDR & Atmos movies on Apple TV (iTunes)
Here is the situation:
Apple TV app support
| Buy/rent iTunes films | 4K | HDR | Dolby Atmos |
---|
Apple TV 4K box |
| | (Dolby Vision) | |
LG TV | | | (Dolby Vision) |
|
Samsung TV | | | (HDR10) | |
Roku TV | | | (HDR10) | |
Fire TV | (can still stream) | | (Dolby Vision) | |