My apologies for bumping an old topic, but I'm having the same problem as the OP with Mint 14 and FF21, and it does not seem to have yet been solved. The original link the OP provided is now updated with instructions for FF21 which is basically the same,
# sudo mv firefox /opt/firefox21
# sudo mv /usr/bin/firefox /usr/bin/firefox-old
# sudo ln -s /opt/firefox21/firefox /usr/bin/firefox
All well and good, but the file named 'firefox' in firefox21 won't execute or do anything. Yes, it's checked as executable. The file 'firefox-bin' doesn't do anything either. Trying to view 'firefox' in gedit throws the error, 'gedit has not been able to detect the character encoding'. Now here's something interesting that may help someone to figure this out: When I go to /usr/bin, 'firefox-old' has one of those terminal icons, meaning executable (I think?), and it starts the current mint version of FF21. However, the new 'firefox' file shows a greyish gears icon (so does firefox-bin). Again, the file is marked as 'Allow executing', and I've tried unchecking and rechecking to no avail. I've not only tried the current version from Mozilla but also the 21beta1 version.
According to the web page that gives these instructions, this is supposed to work -and has been tested- for mint 14 nadia.
I would like the original, non-mint branded version of FF. The PPA and Software Manager both install the mint version.
Thanks for any help