From LifeHacker: Playing Only Audio on iPhone

Rick Betita describes how to stop the video from playing and draining battery on iPhones:
One of the most annoying things about the iPhone is that you can’t listen to the audio of a video file unless the video is playing. Let’s say I’m watching a TEDTalk video podcast where the visuals don’t add much to the content: if I lock my phone or press the home button, playback will stop (rather than continue underneath like an audio file).
To get around it, I lock my phone while I’m watching the video (stopping the playback) and double-tap the home button to bring up playback controls on the locked screen. From there, I can press play and lock the phone again, allowing me to listen to my video file with the screen locked.
If I want to do something else on my iPhone while the audio of the video plays in the background, I press the home button during video playback and lock the screen immediately afterwards. Then I double-tap the home screen as before, hit play, and unlock the phone. It brings me to my home screen while the audio continues to play.