Will painful gums permanently damage me if I don't get treatment? Does it just go away?

Depends on why. Many causes for gum pain. Is it just around one tooth? Is it all of your gums? Do they bleed? Are you taking medication? What's your age? All these factor into answering your question. See your doctor or dentist for appropriate treatment.
It will not. "just go away". It will damage the underlying bone, jeopardize the long term health of your teeth as well as possibly invading the blood stream and causing problem elsewhere in the body. It must be corrected asap. See a dentist or periodontist.