Our Apple TVs continue to lose connection ever since installing Mountain Lion on the host computer. The only thing that fixes them is to quit and relaunch iTunes, which only works unti the ATV goes to sleep. If a movie is paused for more than a few minutes and the ATV goes to sleep, pressing play again on the remote is greeted with "There are no movies on this computer" or "Turn on Home Sharing."
If we stop a video on one ATV and attempt to resume it in another room, the error begins again. The other ATV can't connect until iTunes is relaunched.
My guess is that it has something to do with the new [annoying] security settings. But I still haven't found a fix other than restarting iTunes everytime we want to watch something from our downloads. Fail.