If you try to use HBO Go or Max GO it won't display the video, only audio. I think those premium channels wrote code into those apps to prevent them from being displayed on other devices. Makes sense if you don't subscribe to those services but grab someone's login & password to watch on your TV.
If you have an iPad 2 or a 3rd generation iPad you can airplay anything even if the app doesn't allow it. Just double tap the home button and then slide all the way to the left and you will see that you can mirror what ever you have on your iPad and it will show up on your TV full screen if you have an Apple TV 2nd or 3rd generation,if the show you are watching is in widescreen. You will get The Video with the Audio this way.