- Please verify that the Fire TV system software is up-to-date.
- Verify that the Apple TV app is up-to-date, if not done automatically.
- Verify that your router or firewall doesn’t block Apple authentication servers, in case you made custom changes.
- Verify that your devices and router use a good DNS, like the one from your ISP, which gets updated with all the server names that Apple might use. Again, only applies if you made custom changes.
- Verify that your Fire TV has the correct date, time, and time zone, as Apple may be more strict about that than other content providers (for TLS secure connection reasons).
Else I would suggest to contact an expert from the Apple Support staff for further assistance. Contact Apple for support and service (phone numbers) or start your support request online (chat/email/callback).