Firefox Nightly can be unpacked to the home directory and run from there. However, it's probably a good idea to use a separate profile, to avoid the risk of making changes to an existing profile that are not compatible with the existing version of Firefox (or Iceweasel, in my case).
Debian Bits And Snips explains how to do this. I ran into a problem in that when I'd closed Firefox Nightly and launched Iceweasel again, it was using the new profile. I had to use Profile Manager again to switch Iceweasel back to the default profile, but after that I used the launch command given and Firefox Nightly used its own profile and left Iceweasel using the default profile.
/home/username/firefox/firefox -P "new profile" -no-remoteno-remote means Firefox uses the new profile independently of the profile set for Iceweasel and without changing the profile set for Iceweasel- indeed it can run at the same time as Iceweasel with this command.
Profile manager can be launched with the following terminal command in Linux, if Iceweasel is using the wrong profile: