Tools > Addons > Extensions

What extensions do you have installed?

The Web Developer extension (and probably other extensions) has a preference to show the Error Console when errors are detected. Deselect that pref in Options for that extension.
http://chrispederick.com/forums/

Found this solution in http://www.velocityreviews.com/forums/t590971-how-do-i-go-about-disabling-the-error-console-in-firefox.html and it worked for me (for Firefox 5.0):

"I've been trying to get rid of the error console popping up all the time for some time now - it would open every time I opened Firefox, considerably slowing down startup. I disabled all of my add-ons and re-enabled them until I came upon the problem. I found out that I had two different mouse-gestures add-ons installed, causing a conflict which made it open all the time. I'd recommend disabling all of your addons and re-enabling them one by one just to check if that's the problem. " (posted by willeifler)

In my case, the problem was with "FireGestures" and "All-in-one Gestures". Disabling one of them solved the problem.

Thanks to him/her. I hope this will help you, too!