From MozillaZine Knowledge Base
Firefox may report that an update is available even though you are running the most recent version of Firefox and no updates exist. One of these methods should fix the problem.
Close Firefox and delete "active-update.xml", "updates.xml", and the "updates" folder from the folder where Firefox stores temporary application data for your operating system.   Important: This folder may be hidden by the operating system. Read Show hidden files and folders.
For example, on Windows 7, the "active-update.xml", "updates.xml", and the "updates" folder can be found in one of these locations :
- C:\Users\<username>\AppData\Local\Mozilla\Firefox\Mozilla Firefox
- C:\Users\<username>\AppData\Local\Mozilla\updates\<alphanumeric string>
Download the latest version of the Firefox setup file from www.mozilla.org/firefox/new and install it "over the top" of your current Firefox installation.  
Software Update - Software Update not working properly
Related bug reports:
- Bug 368661 – on install, remove the updates directory, updates.xml and the active-update.xml file
- Bug 372981 – Update history missing after update to 126.96.36.199
- Bug 374140 – Software Update (Firefox 2001 -> 2002) Problem - wrong User Agents
- Bug 374900 - Update without permissions fails - then gets confused after administrator update
- Bug 375710 – Firefox wants to downgrade to 188.8.131.52 from 184.108.40.206
- Bug 376305 – scenario where the user could get prompted to update to 220.127.116.11 (even though they have 18.104.22.168)
- Bug 572162 - Remove requirement for app dir to be under program files for UAC elevation (breaks side by side installs of x86 and x64 - mixing of updates including update history)