Body Mushroom (literally meaning "Mushrooms Growing from the Body Shroom") is a species of mushroom that grows on Amazon Lily.
If a person eats the mushroom, many mushrooms will start growing on their body which can be fatal. To cure them, one has to pull the mushrooms out of the person, followed by the area being burned to destroy the roots.
Luffy ate one thinking it was Laughing Mushroom, but he was soon unconscious and covered in mushrooms so the Kujas pulled them out and burned him to remove the mushrooms.
The medical treatment of "burning" the mushroom to get it off is indeed a natural cure for several biological diseases (notably fungi such as the verruca) that grow on the skin. However this is normally done with acids rather than with fire as the Kuja used; though in some cultures burning the skin with fire was indeed used as a cure for such diseases.
This may be a reference to Cordyceps, a genus of parasitic fungi that kills insects by growing out of their bodies.