First quit all currently running instances of Firefox:
killall firefox
Then download the AppImage from https://bintray.com/probono/AppImages/Firefox/56.0.2.glibc2.3.4#files
mkdir Firefox-56.0.2.glibc2.3.4-x86_64.AppImage.home
./Firefox-56.0.2.glibc2.3.4-x86_64.AppImage
After running the AppImage, you can inspect the contents of Firefox-56.0.2.glibc2.3.4-x86_64.AppImage.home
and you will see stuff has been saved there:
me@host:~/Downloads$ find Firefox-56.0.2.glibc2.3.4-x86_64.AppImage.home/ | head -n 10
Firefox-56.0.2.glibc2.3.4-x86_64.AppImage.home/
Firefox-56.0.2.glibc2.3.4-x86_64.AppImage.home/Desktop
Firefox-56.0.2.glibc2.3.4-x86_64.AppImage.home/.cache
Firefox-56.0.2.glibc2.3.4-x86_64.AppImage.home/.cache/fontconfig
(...)
Firefox-56.0.2.glibc2.3.4-x86_64.AppImage.home/.cache/fontconfig/CACHEDIR.TAG
Firefox-56.0.2.glibc2.3.4-x86_64.AppImage.home/.cache/mozilla
(...)