Hello, sounds like your're sending the wrong useragent string to Youtube. Go to http://whatsmyuseragent.com/. If it says something about "MSIE" instead of Tenfourfox/Firefox, you can reset the useragent in about:config. Right-click general.useragent.override and choose reset.
Then I'm out of ideas, I'm sorry. I can't reproduce the behavior you described when the browser sends our default useragent string. Or maybe you have some site specific override for Youtube defined (by an addon)? This should then show up in about:config as well. general.useragent.site_specific_overrides = true is default and does not cause this behavior for me.
I was having the same problem and today, Youtube refused to let me watch anything. My problem was due to the add-on "ALL HTML5" that was altering the useragent string. Disabling this solved the problem.