Based on my experience, NBA League Pass is not compatible with Fire TV. I signed up for the free trial of League Pass on Tuesday, October 27, 2015, and as of the afternoon of Friday, October 30, 2015, I have not been able to watch a single game utilizing the app on my Fire TV.
Further, and perhaps more disappointingly, I have expended far too much effort attempting to resolve this issue with NBA League Pass support ( including one representative explaining that I might not be able to watch the Atlanta/Detroit game from my home in Seattle due to local blackout restrictions) . This has been an incredibly frustrating experience.