Is it possible to run selenium (Firefox) web driver without a GUI?

What you're looking for is a headless-browser.

Yes, it's possible to run Selenium on Firefox headlessly. Here is a post you can follow.

Here is the summary steps to set up Xvfb

#install Xvfb
sudo apt-get install xvfb

#set display number to :99
Xvfb :99 -ac &
export DISPLAY=:99    

#you are now having an X display by Xvfb

Chrome now has a headless mode:

op = webdriver.ChromeOptions()
op.add_argument('--headless')
driver = webdriver.Chrome(options=op)

Yes. You can use HTMLUnitDriver instead for FirefoxDriver while starting webdriver. This is headless browser setup. Details can be found here.

If you want headless browser support then there is another approach you might adopt.

https://github.com/detro/ghostdriver

It was announced during Selenium Conference and it is still in development. It uses PhantomJS as the browser and is much better than HTMLUnitDriver, there are no screenshots yet, but as it is still in active development.