I've noticed since iOS 10's public release, when I airplay to my Apple TV, and lock the screen, the audio seems to occasionally stop. I've gone on to iOS 10.1 public beta, this behaviour remains. I know some people are having Bluetooth issues, not 100% sure these are connected, as I have had no problems with my Bluetooth headphones over the last week and a bit.
The only issue I've had with Bluetooth is some of my bt devices won't pair (such as my foxL v2 platinum). Just doesn't show up. Never had that issue on my 6 Plus. May be a bug in the iOS 10.1 beta I installed earlier today.
See, I've had no Bluetooth issues. I've seen others have, so I'm not sure if my airplay issue is connected.
Maybe it's the same problem as it was in ios9. If you had a live wallpaper on the lockscreen the music stops when the screen goes black.