How can you listen to Youtube videos on iOS, while the phone sleeps?
Step 1. Launch Safari, the default iOS web browser.

Step 2. Search for YouTube in the address bar / search box, or simply type youtube.com and search for a video that you want to listen to.
How to listen to YouTube in the background in iOS 7

Step 3. Let the video begin to play - wait until any advert has finished and the actual video is playing.

Step 4. Press your device's home button to return to the home screen. The music will stop. But, swipe up from the bottom to display Control Center. Tap the play button to continue playing the audio from the YouTube video.
In later versions of iOS, you may have to swipe between tabs in the Control Center to see the playback controls.
Play YouTube in background iOS 7 - Control Center

Step 5. You can now launch another app, such as Messages or Mail and keep listening while using that app. Some games (those which allow you to keep listening to your own music) will still be usable without cutting off the audio from the YouTube video.
If you want to browse the web, you'll need to use a different web browser from the one which is playing the YouTube video, since switching tabs will cause playback to stop. If you don't have another browser installed, try Google Chrome, Firefox or Opera.

Step 6. You can listen to YouTube with the screen off, too. Press the sleep/wake button to turn your device off and the audio from Safari will stop. Now press it again and tap the play button on the lock screen to restart the audio. Now when you turn off your iPhone or iPad, the audio will keep playing.