We're having the same problem but not related to iOS7 as suddenly my husband's iphone 5 and ipad 3 both won't let him choose the AppleTV. It's like it doesn't "see" the AppleTV is there anymore. My devices work fine in the same household.
This was sudden. This afternoon. Really would like some suggestions on how to fix this. Have tried rebooting, router & restarting & updating ATV3.