While it's not a Firefox add-on, rather a command line tool, I can highly recommend the open source youtube-dl.Can anybody tell me if there is a safe Firefox add-on for downloading non-copyrighted YouTube videos?
Probably the hardest part for some folks would be installing it. Instructions can be found in the README.md file. After that, it's very simple to copy the URL of the page containing the video in Firefox and paste it into a command line in Terminal. youtube-dl can even automatically download all the videos in a playlist.
You bring up the issue of copyright infringement. If YouTube plays a video without being logged-in, then youtube-dl will be able to download it as well. This does not preclude people from uploading things to YouTube that should not be there for reasons of copyright. It is up to the copyright holder to ask YouTube to remove a video if it is unauthorized.