[UPDATE 11/6/2009: Today, I upgraded Alexa Sparky to Version 1.4.5 and re-enabled it. If there are no problems with the FireFox toolbar, it means that probably, Alexa fixed the problem and you can go ahead and do the same, update Alexa Sparky to 1.4.5 or higher and enable it. Let me know in a comment if that worked for you. I will keep you posted as well.]
[SOLVED: This is a quick, 2 minute solution. I found out that Alexa Sparky Version 1.4.4 Firefox Extension/Add On which I downloaded today was the culprit. Do you have Alexa Sparky AddOn? I disabled Alexa Sparky 1.4.4. under Tools/AddOns and the address bar in Firefox 3.5.3 works again. See details below to make sure your problem gets solved this way too!]
Previously, I reported blinking Close, Minimize and Maximize button in a Firefox 3.5.3 frame using Windows Vista SP3 . Shortly after I observed that problem, the address bar functionality has been severed. I can not navigate to any websites, not even google.com. I can not see the URL of any website in various tabs that I have open. That is true even after restarting Firefox. I am about to downgrade to Firefox 3.0.14. I've downloaded it and ready. I got importan work to do online! And Firefox is my main window to the internet. If you have found any solution to this, please let me know. I could not find a solution on the internet as of yet.
SOLUTION (SAME DAY IN THE EVENING)
I found out that Alexa Sparky Version 1.4.4 Firefox Extension/Add On which I downloaded today was the culprit for the problem with the address bar described above. I went ahead, disabled Alexa Sparky under Tools/AddOns and the fabulous address bar in Firefox 3.5.3 works again.
P.S. Before downgrading to 3.0.14. I will still see if closing some of the Firefox tabs might help. I have 2 windows open with Firefox with about a dozen tabs each.
P.P.S. For future benefit, I'll post a link to Mozillazine.org's Standard Firefox Diagnostics and Troubleshooting process. A bit over the top for this case. Especially as I wanted to retain the valuable tabs opened. Here is the link: http://kb.mozillazine.org/Standard_diagnostic_%28Firefox%29