Some programs work great on my Apple TV and others give me an error: there is a problem loading this video. If this continues, check your internet connection.

Something is causing the app to be unable to contact the remote servers.

See if this document has clues that apply to your situation: If you can't watch or stream movies and TV shows in the Apple TV app - Apple Support

“Video not available” may mean no valid reaction from the remote server.

1. Verify that your router doesn’t block Apple’s servers (or its CDN) through a firewall setting.
2. Verify that your router doesn’t block ports that Apple services like to use.
3. Verify that you use a good DNS on your router, hopefully the one from your ISP, which gets updated with all server names that Apple might use.
4. Verify that your Roku has the correct date and time and timezone, as Apple may be more strict about that (TLS) than other content providers.

Else

1. Check for Roku OS software updates.
2. Check for Apple TV app updates, if not set to do so automatically.
3. Try resetting the Apple TV app.
4. Try deleting and re-installing the Apple TV app.
5. Try restarting the Roku.

If the issue persists after all that, then 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).