Recently, presumably after an automatic software upgrade, my Fire TV Cube continues to pay the last program's audio after switching off the device from the remote control. The only way to stop the audio is to unplug the Fire TV Cube's power cord.
I'm sorry you're having this issue. So I can do some research and get the best advice for you, I need a few more details. Do you have a receiver/soundbar or external speakers connected to your Fire TV Cube? Is the audio coming through those or through the Fire TV Cube itself?