Listening to iTunes U audio in the background
I understand your frustration, thankfully, it can be done.

Many video apps, like the YouTube app or the iTunes U App allow you to play the content in the background like music, it just takes a certain trick.  The way you do it is two ways, depending on if you want to sleep your iPhone or use other apps while you listen:

Exit the App and then double click the home button, and then swipe to the right to reveal the iPod controls, you should see the youtube or iTunes U app in the place of the iPod Music Icon.  Hitting play should continue playing the contents audio in the background while you do other things. Opening another music or sound related app may mute or stop the playing but opening up Safari or checking your mail, text messages, etc should not stop anything.

To have it play in the background with your iPhone in standby mode, i.e., like listening to your music while your iPhone is sleeping. Simply lock your iPhone, the sound will stop. Double click the home button and the iPod controls will show up, press the play button and the audio portion of your iTunes U or Youtube app content should keep playing.  

I hope that this information helps you out.