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.