I had ovarian cysts. Well, I still do have them but they're much smaller now due to my taking birth control. They would most likely have a laser or something like that and nothing really invasive, since it's right in your mouth. They hurt like HELL when they explode. One of my ovarian cysts exploded during my algebra 2 final when I was a freshman. It would probably feel a bit different in your mouth, but imagine a small pellet full of tiny razors in your gut exploding. I was in so much pain I passed out.
Whether or not it will go away on its own or not depends on what it's filled with. Since cysts aren't tissu like cancerous tumors are, they are filled with liquid. Mine were filled with testosterone so I went on birth control to level it out with the female hormone. I've never had any of mine invasively removed, but I don't imagine it would be painful, since you would be under anesthesia.
If you need any more info feel free to message/email me!