Personally, if it is not huge in size, not hard to get to, you have no fever, rash, shortness of breath, N/V, headache, back or joint pains, you can eat and drink normally, you don't feel "sick," you don't have any underlying medical problems, you can easily get to it, and you have no problem with sharp objects, THEN you MIGHT try to drain it yourself with a needle or small, sharp knife. I KNOW I would try this FIRST, but I am a doctor and self-draining a boil is very acceptable. Of course, this may sound like medical heresy as I know the medical mainstream will tell you to go to your doctor -- and doing that would almost NEVER be wrong! So, if you already have a doctor, I am definitely on board with that if it is not a problem for you, you can quickly get in, and you have health insurance (or don't mind paying cash if you don't), etc.
But, IF it is large, hard to get to, you have a lot of pain and/or fever and/or you don't feel well, then you could also go to an "URGENT CARE CENTER" for I&D or even an ED for I&D (incision and drainage) if you are really sick. But you would probably be in for a long wait!
Once a boil is completely drained and then lightly washed with soap and water daily (and you have no underlying disease that could predispose you to infection and/or complications like diabetes or AIDS, or if you have some other underlying immune deficiency), it will heal quickly as MOUTH lesions do heal more quickly than other areas of the body!