Place pork roast in a slow cooker, rub with salt, and pour root beer, liquid smoke, and garlic over the top.
Cook on Low for 8 to 10 hours. Remove pork to a bowl. Strain liquid through a fine-mesh strainer; discard liquid or save for another use (see Editor's Note). Add garlic cloves back to to pork. Shred pork using two forks. Serve with your favorite BBQ sauce.