1. Create a temporary folder for a temporary profile,
  2. Run Firefox with argument -profile path/to/temporary-folder -no-remote -new-instance
  3. You may delete a temporary profile after you close the Firefox instance.

For windows, To run firefox with the argument, at the command prompt, type the following:

> start firefox -profile c:\path\to\temporary-folder -no-remote -new-instance

Are you a linux/unix user? If so, you can easily do it with the following shell script:

#!/bin/sh
PROFILEDIR=/tmp/tmp-fx-profile.$$.d

mkdir $PROFILEDIR
firefox -profile $PROFILEDIR -no-remote -new-instance
rm -rf $PROFILEDIR

According to the Sangye's comment, using mktemp(1) is more better:

#!/bin/sh
PROFILEDIR=`mktemp -p /tmp -d tmp-fx-profile.XXXXXX.d`
firefox -profile $PROFILEDIR -no-remote -new-instance
rm -rf $PROFILEDIR