Luckily, the solution is fairly straightforward.  Once you have SSH’d into a remote host (using ssh -X), you simply need to run firefox -no-remote. Of course you may want to tack on > /dev/null and an ampersand & to ignore the output and background the task. (Thanks to The Open Sourcer.)

With Firefox 2.x this behaviour was somewhat undocumented, but with Firefox 3.x, running firefox --help from the command line shows the -no-remote option.  It also seems that the default (i.e. -remote), is “documented” on Mozilla’s site for Remote Control of UNIX Mozilla.

If you wanted to make the -no-remote behaviour the default when SSH’d into remote machines, you could simply add a few lines to your bash profile to alias the firefox command.

# If we're forwarding X over SSH, make firefox execute on this machine
if [ -n "$SSH_CONNECTION" -a -n "$DISPLAY" ]; then
    alias firefox='firefox -no-remote'
fi

At least that is what I did.