How can I play YouTube in the background without using Safari?
On iOS 5.1 (not sure for earlier versions) the same procedure works for the YouTube app too (as for Safari): 


Start the video in the YouTube app.
Press "Home" button once to go to springboard (or twice to switch to another app)
Press "Home" button twice to bring up the app-switcher. Slide to the right to bring up background audio controls. The icon on the right side will show whichever program you last used with background audio: YouTube.
Press Play. YouTube audio will resume playback.