Went with this extension first since it is recommended by Mozilla, but it didn't work for me at all on my Motorola phone. After trying many similar ones in the add-ons store, the best I've found so far is:
1. Background Playback Helper (YouTube & Vimeo) by phase3
2. Background Video Player by Dr. B. R. Shriradhakrishnasharnam

Hopefully this helps someone save some time. This is an older extension and hasn't been updated in more than a year and a half despite YouTube's recent changes to target non-premium users. There appear to be many newer ones, like those above, that work better based on my testing.

Recently it breaks quite substantially youtube: it everytime I reopen the site while a video is playing, it's in landscape. Also when I pass from a page to another

Great, great, great! Thank you, thank you, thank you! My privacy is important. You make Mozilla Firefox even greater with your presence.

Perfect for YouTube music

God this extension made all my phone freezing and playback randomly stopping when i open another app, horrible experience, once i uninstalled this extension browser became faster. I dont even know that one extension can cause so much trouble

Not working with the screen locked for me. Next video will attempt to play for about 2 seconds, then stop until I unlock the phone again.

Excelente extensión y muy necesaria actualmente.

Ok gass

It stops after a random duration, sometime 10 minutes, sometime not even 1. Very annoying when you listen to a music playlist

Excellent, allows to listen with the screen locked