Do you see a tooltip if you hover that icon?

Start Firefox in Diagnose Firefox issues using Troubleshoot Mode to check if one of the extensions or if hardware acceleration is causing the problem (switch to the DEFAULT theme: Firefox (Tools) > Add-ons > Appearance/Themes).

• Don't make any changes on the Safe mode start window.
• https://support.mozilla.org/kb/Safe+Mode

No, nothing happens at all. It is simply an icon of a TV screen with the old style set-top antennae and blue screen. No reaction to right or left click nor to hover over.

Any ideas?

That icon isn't a default icon that comes with Firefox. Something you installed added that icon to Firefox and you'll have to figure out which add-on that icon is for. No tooltip means that you'll need to disable add-ons until it disappears and then you'll know which add-on that icon is related to.

http://support.mozilla.com/en-US/kb/troubleshooting+extensions+and+themes