I just got firefox working with xbmc live and wanted to write it up because I learned some things along the way. This worked for me but is not guaranteed to work for everyone. There may be better ways to do this that I am unaware of.
1. Get firefox. Enter terminal mode with ctrl-alt F3
sudo apt-get install firefox
I had to do sudo apt-get install update before I could download firefox
2. Make a shortcut directory that xbmc can see
sudo mkdir /home/(your username)/shortcuts
3. make a symlink to this directory for firefox that allows you to run firefox from launcher
ln -s /usr/lib/firefox /home/(your username)/shortcuts/firefox
4. Install launcher
get it from here
http://code.google.com/p/xbmc-launcher/downloads/list
use winscp to put in /home/(your username)/.xbmc/plugins/programs
5. Reboot to xbmc and open programs, program plugins, launcher
6. Choose standalone (normal PC executable) and choose firefox from Home folder/shortcuts
7. At application arguments you can hit done. Set title of the launcher as firefox. You can get a firefox thumbnail and add to favorites if you desire.
8. When you open firefox, it may only open in half the screen. You get mouse and keyboard focus by hitting the \ key. You can resize it by dragging or hit f11 for fullscreen toggle. Once you exit firefox you will need to hit the \ key to once again see xbmc fullscreen.
9. One other thing. Installing firefox somehow will ruin your suspend/awake from suspend control by your remote (if you have this working). There is a ticket for this. Basically, under shut down functions, if you see ?0? and ?1? for choices, then you have this issue. For me the following solved it.
sudo apt-get remove policykit-1
sudo apt-get remove devicekit-power
You can see this thread
http://forum.xbmc.org/showthread.php?tid=65726 for further details regarding this.
Hope this helps.