You can upgrade to Firefox 29 beta that is available in the Mozilla team PPA.

To install Firefox 29, run the following commands in the Terminal:

sudo add-apt-repository ppa:mozillateam/firefox-next && sudo apt-get update

After these commands you will see the text:

You are about to add the following PPA to your system:
 This PPA contains releases of Firefox from the beta channel

All users of this PPA are encouraged to report any bugs they find. Instructions for reporting bugs can be found at https://wiki.ubuntu.com/MozillaTeam/Bugs. Please have a read of this before reporting a bug, as it may help save time.

Please don't email the package uploader to report bugs. Not only is email not a good tool for tracking bugs, it also excludes anybody else from tracking or working on the issue.

For questions and bugs with software in this PPA please *don't* contact mozillateam.
 More info: https://launchpad.net/~mozillateam/+archive/firefox-next
Press [ENTER] to continue or ctrl-c to cancel adding it

Press ENTER and wait for the message:

Reading package lists... Done

Then run the command

sudo apt-get install firefox

and press y when you see the message:

Do you want to continue [Y/n]?

When the installation is finished, you will see the message:

Please restart all running instances of firefox, or you will experience problems.

Now you can start using Firefox 29 beta by launching it from the Mint menu.

You can find it in the "Internet" category under the name "Firefox Web Browser", as it replaces your previous installed stable version.

You can also install Firefox 31 alpha that is available in the Mozilla daily PPA.

To install Firefox 31, run the following commands in the Terminal:

sudo add-apt-repository ppa:ubuntu-mozilla-daily/ppa && sudo apt-get update

After these commands you will see the text:

You are about to add the following PPA to your system:

 daily (or even multiple builds per day) for various mozilla projects and branches.

 

For questions and bugs with software in this archive, please contact

 More info: https://launchpad.net/~ubuntu-mozilla-daily/+archive/ppa

Press [ENTER] to continue or ctrl-c to cancel adding it

sudo apt-get install firefox-trunk

and press y when you see the message:

Do you want to continue [Y/n]?

When the installation is finished, you will see the message:

Please restart all running instances of firefox-trunk, or you will experience problems.

Setting up firefox-trunk-globalmenu (20.0~a1~hg20130107r117821-0ubuntu1~umd1~quantal) ...

Setting up xul-ext-ubufox (2.6-0ubuntu0.12.10.1) ...

Please note that the name firefox-trunk is used instead of firefox as it will be installed as "Nightly Web Browser".

You can start using Firefox 31 alpha by launching it from the Mint menu and it should work at the same time with your current stable or beta version.

You can find it in the "Internet" category under the name "Nightly Web Browser".