I've set "allowsAirPlay" to "YES" in my MPMoviePlayerController, but AirPlay button doesn't appear. Does iOS simulator support AirPlay? And give please some example how to stream video through Apple TV for example.
Nope, at the time of writing (iOS 7 just released) simulators don't have airplay nor any way to simulate or test it. Having an airplay capable device (apple TV) or program (http://www.airsquirrels.com/reflector/) in the same wifi network makes the mirroring button appear on my iPhone 4S but not on any simulator.
Edit: There is a way of testing it, simulate TV output.