Have you looked in Chromecast help for compatibility with Firefox? Or lack thereof? https://support.google.com/chromecast/?hl=en#topic=6314574

Try to toggle browser.casting.enabled to true on the about:config page.

You can open the about:config page via the location/address bar. You can accept the warning and click "I'll be careful" to continue.

• http://kb.mozillazine.org/about:config

Found something on Reddit: https://www.reddit.com/r/Chromecast/comments/2rhmk0/any_way_to_cast_from_firefox_yet/

It seems that Google is not willing to share casting in other browsers then Chrome. Maybe someday.... For now I'm mirroring my browser on TV through a "good old" HDMI cable.