As far as I know, Firefox still allows you to play video in the background from the lock screen. It's probably YouTube that changed (because they want you to pay them for the feature, because people use YouTube for background music).
You can keep it from pausing by requesting the desktop site in Firefox. The mobile site pauses, the desktop site just keeps playing.
Interesting. I tried it in Firefox for Android, and the desktop site plays for a couple of seconds after screen lock and then stops. It might depend on the video.
It probably also depends on the power saving settings your phone enforces. From what I've seen from DontKillMyApp[0] the optimalizations vary widely per manufacturer and probably per device and ROM as well.
I don't think the websites cares much, at least it didn't for me and I tested it with a music video (something Youtube usually doesn't like you playing in the background without a subscription). Your phone's settings and optimizations are a more likely culprit, in my opinion.
You can keep it from pausing by requesting the desktop site in Firefox. The mobile site pauses, the desktop site just keeps playing.