I was having this same problem in Linux Mint Maya (based on Ubuntu 12.04): Banshee is playing, monitor powers off after some amount of idle time (though screen does not lock), Banshee finishes playing track and then stops. When I move the mouse and the monitor comes back on, Banshee starts up again.
I wondered why Banshee was going to the end of the song and THEN stopping. So I figured maybe it had to do with new-track notifications. Sure enough, I disabled Banshee's "Notification Area Icon" extension (In Edit->Preferences->Extensions tab) and now Banshee keeps on playing just fine, new tracks and all, after the monitor goes off. I don't get pop-up notifications between songs anymore, but that's a small price to pay on my end.
I'm going to go mention this in a related bug report; if it works for you, consider saying so there so that we might help the developers find the real source of this issue.