Several Firestick and FireTV users have commented that when they leave an app (normally a music or TV app) the sound continues to play. Here is the Amazon recommended method for resolving the issue.
First we’ll want to restart the Fire TV fully. Do this from the power button or by unplugging and re-plugging in the power.
Next let’s clear the Cache and App data by going to Settings, then click Apps, go to Manage Installed Applications and choose Amazon Music. You’ll see a “Force Stop” option to choose on this screen; this will force stop all app activity, such as playing music.
To help in the future you can also disable auto-play from the home screen by scrolling to Settings, click on Preferences, then scroll and select Featured Content and turn off both “Allow Video to Play” and “Allow Audio to Play”.