Hello Jaeknj,
Thank you for reaching out to Apple Support Communities.
Let's delete the Apple TV app and reinstall it one more time using these steps:
Add the Apple TV app from Amazon.comOptionally, you can sign in to Amazon.com from a web browser to add the Apple TV app to your account.For a list of specific Fire TV models that support the Apple TV app, go to apple.com/apple-tv-app/devices/.1. In a web browser, go to the Apple TV App in Amazon.com, then sign in to your account, if you haven’t already.2. Follow the onscreen instructions to select the Amazon Fire TV device on which you want the app installed, then select Deliver.
When added, the Apple TV app appears in the Apps & Channels section of the home screen on your Fire TV.3. To start watching, see How to watch on your Fire TV above. To learn more about what you can do in the Apple TV app, see the Apple TV App User Guide for Smart TVs and Streaming Devices.
This article can provide more details: Watch on Amazon Fire TV
If the issue persists, can you provide a few more details? Hav you tested on another network? Do you know if your Fire TV is up to date? Also, when did you first notice this?
Good Luck