Warning: Long explanation for something that's relatively easy to do. It's also a "beginners" explanation--I can't help it, I used to write tech manuals.
This is rather easy to do in LMDE once you know how, and it's due to Debian and Debian based distros having the actual application like Firefox, Thunderbird, etc installed in the /opt/ directory rather than in various places throughout the system as Ubuntu does.
Note: Your Firefox profile is still stored in your Home directory in the ".mozilla" directory as it is in Ubuntu and Linux Mint main edition and will not be touched during this procedure.
First you need to download the Firefox ".tar.bz2" file from Mozilla. Since you're running the 32 bit version of LMDE you can download directly from the Firefox download site itself.
http://www.mozilla.org/en-US/firefox/new/
However, just for information sake, if you're running the 64 bit version of LMDE you'll need to download the 64 bit version of Firefox from Mozilla's FTP site (direct link to version 20 FTP directory):
http://ftp.mozilla.org/pub/mozilla.org/ ... ases/20.0/
Make sure you click on the "linux-x86_64/" directory, then you can download the 64 bit version in your language.
Once the file is downloaded, close Firefox.
Next, open your file manager, select "File system" from the left sidebar. Now right-click the /opt/directory and select
Open as administrator. In the new (root) file manager window (the /opt/ directory), you'll see both Firefox and Thunderbird folders. Since you just want the latest Firefox, rename the current Firefox folder to "firefox.bu", "firefox.orig" or some such thing (always back up the original install). Now, in the first file manager window, find your downloaded Firefox ".tar.bz2" file and drag it into the /opt/ directory window. Once the file is copied into /opt/, right-click it and select
"Extract here" from the menu. Once the extraction is complete you should have a new "firefox" folder in /opt/.
At this point just start Firefox as you normally would and if all went well, you're latest version of Firefox should come up, check add-ons for compatibility and bring up the browser. If so, then
Delete (not "Send to trash") the .tar.bz2 file from /opt/, close down the file manager windows and you're done.
To update Firefox at this point, just open Firefox and go to "Help/About". If it shows that an update is available, close Firefox, open a terminal and type:
Firefox needs root access in order to update
Once Firefox comes up again (it will appear as if you just installed it...no add-ons), navigate back to "Help/About" and click the update button if it doesn't start updating automatically. Once the update is complete it should ask you to restart the browser, do so. Once the browser restarts, close it back down (and you'll see the terminal return to the command prompt). Close the terminal and you can now start Firefox normally.
Okay, as I said, long explanation for something that's easy to do once you know. One caveat is that once the "official" upgrade comes through the repositories, it will happily overwrite the "firefox" folder in /opt/. This really won't affect you until the next version of Firefox is released then you'll have to repeat the above procedure. Or, you can just "Ignore" the "official" upgrade via MintUpdate and continue upgrading Firefox within the browser itself.
Hope this works for you.