If you wish to remove the hole, you would need to have it excised and sutured up. This can be done with local anesthesia. I would recommend seeing a surgical dermatologist (Mohs surgeon) or plastic surgeon.
You have two options: you can keep the hole in your lip or you can choose to excise it (have it cut out) which would leave you with a small scar in place of the hole. Excising the hole could be done under local anesthesia in an office setting.