Firefox may be the problem since the video is working correctly on your phone.
I suggest trying to:
1) Clear your entire cache and history, sign back on to facebook and see if the video plays.
2) Make sure your firefox is updated to the latest version. If not, install it.
3) Make sure you are running the latest version of Flash Player. You can download Adobe flash player from the download section here: http://ccm.net/download/download-91-adobe-flash-player
4) Restart your computer and then refresh your browser.
If all else fails, test Facebook videos on Internet Explorer or Google Chrome and see if they are working.