You would need to pretend to be desktop Firefox. You can do this by clicking on the menu button and selecting request desktop site.

But surely there is also a way to view videos in the regular mobile site without using the app? (As in: I want the same behaviour in Firefox as there would be if I did not have the app installed.)

No. If the YouTube app is not present the site will fail to work. The only known work around is to use the desktop YouTube site.

It depends on your phone but on my galaxy I can go to the settings and set the default app to firefox or whatever other app is able to play a video. What device are you using?

Thanks kbrosnan, I'm afraid that's that then.

@jjsalzano I'm using a Fairphone (Android 4.2), but I can find no such option in either YouTube's, Firefox's or the system settings.