Diagnosis
This is a common Fire TV OS software glitch, often introduced or worsened by a system update. The audio processing is getting delayed relative to the video.
How to Fix It
Go to Settings > Display & Sounds > Audio. First, try switching the 'Audio Output' setting between 'PCM' and 'Auto'. If that doesn't work, look for an 'AV Sync' or 'Audio Delay' setting and adjust it manually until the lip sync matches.
If That Didn't Work
Perform a full 'Clear Cache' and 'Force Stop' on the specific app having the issue (like Prime Video) via Settings > Applications > Manage Installed Applications. Then restart the TV.
Heads Up
If you're using an external soundbar or AV receiver via HDMI ARC, the sync problem might be there. Try using the TV's built-in speakers to see if the problem persists—if it does, it's the TV's fault.