Is there a way to play YouTube videos in the background? I remember that once upon a time I was able to start a song on the YouTube app (or Jasmine), then go and do other things on the phone.

That doesn't seem possible anymore with iOS 10.  Am I missing something simple?
Yes! You only need to enter from safari and not from the app. You choose the song you want to listen to, press the home button, swipe up to multitask and press play!