Go to Tools | Options | Tabs and check to see whether "Always show the tab bar" is checkmarked or not.

I'm having the same issue, I tried turning off the 'Always show the tab bar' option as well as the 'Show tab previews in the Windows taskbar' option, but this did not help. I also tried switching between Windows Classic & Windows Aero themes, no dice.

I just moved from Windows XP to Windows 7. On my old XP box, it is working fine. But I can't add the 'Open a new tab' button on my new Windows 7 Pro x64 system.

You can find the New Tab button showing as a '+' on the Tab bar.
You can open the Customize window and drag the New Tab button from the Tab bar on another toolbar and it will become a regular toolbar button like the New Tab button that you have in Firefox 3 versions.
If you want the New Tab button at the far right end of the Tab bar then place a flexible to the left of it.

• If items are missing then open the Customize window via "View > Toolbars > Customize" or via "Firefox > Options > Toolbar Layout"
  
• If a missing item is in the toolbar palette then drag it back from the Customize window on the toolbar
  
• If you do not see an item on a toolbar and in the toolbar palette then click the "Restore Default Set" button to restore the default toolbar set up.

See also:

• http://kb.mozillazine.org/Corrupt_localstore.rdf