I'm not aware of any third-party receiver that supports video over AirPlay, so you'll probably need an Apple TV. But then, as long as you have a WiFi network in your house, you should be able to play the movie through the Apple TV to your projector without needing Internet as long as the movie has completely downloaded. For more information on AirPlay see:
http://support.apple.com/kb/HT4437
In some cases, though, people have reported that even with the rental completely downloaded (or so they said) it still asked to connect to the iTunes Store, so it's not 100% reliable, it seems.
Regards.