Hmmm - working fine for me (iPhone 5 here) - I've a few movies in iTunes (that I've manually converted to .m4v using iVI Pro, these are not films from the iTunes Store); if I drag one to my iPhone and then go into the "Videos" app, I can see them just fine, I select one and it has the movie info plus a play button, hitting play takes me into the normal "playback" window, and there is the AirPlay button in the bottom right-hand corner, I can tap that and send the video output to my Apple TV, which is exactly what I would have expected?
What format are the video files you are trying to play?