Right click a blank part of the Tab Bar and then click "Add-on bar" to remove the checkmark.

Incidentally, some of your Plugins are seriously out of date which exposes your system to attack. Please visit the Plugins Check page and update where necessary.

Thanks Xircal but I've done that and the add-on bar keeps coming back. I've clicked on the red X in the upper left hand corner, it keeps coming back. I've gone to the "Options" menu and unchecked "Add-On Bar" but it keeps coming back. Any other suggestions?

Some add-ons can cause the add-ons bar to re-appear, for example the McAfee Site Advisor extension can cause the add-on bar to reappear even if you hide it.

To find out which add-on is causing this for you, follow the procedure in the following link- https://support.mozilla.com/kb/Troubleshooting+extensions+and+themes

TonyE - I followed your link above, and sure enough I disabled McAfee Site Advisor and now the add-on bar stays off. I guess this means I need to choose which I want - McAfee or no add-on bar?

I think for me the problem was caused by Adblock Plus. Other then Zotero, that's the only add-on I have visible there. I set the option Adblock Plus options"Show in Toolbar" and "Show in Statusbar" off, and am still hoping that did the trick.