Can a puncture wound in your mouth heal?

Natural Healing. The good news is that area has fantastic blood supply and so healing will occur rapidly. Inflammation (red, hot, tender swelling) is the body increasing the blood flow to the injury to bring healing factors to the area and also the veins expand to carry bad stuff away. Dressings and ointments are of no value there. Avoid further trauma to the area and let nature take it's course.
Yes. Mouth puncture wounds will heal once the cause is treated. Wound bed need to be cleared of infection due to puncture so once there is no infection, inflammation tissue will heal. To accelerate healing use orally riboflavin - vitamin b2 in a higher dose and methionine - together these two supplements will help mouth lesions heal quicker.