It is possible that the piercing could be removed and repaired. The skin that grew down into the piercing would need to be removed to allow the hole to heal. This does result in a small scar but in most cases the scar is less noticeable than the initial piercing hole. Laser resurfacing or dermabrasion can be used to further camouflage scar tissue. I hope this information is helpful for you.
Stephen Weber MD, FACS
Denver Facial Plastic Surgeon
Piercing holes can be repaired using minor surgery to remove the epithelialized tract and realign skin edges.
Raffy Karamanoukian MD FACS
It is possible that the hole on your lip from a past piercing could be surgically removed. Photos or an examination would be needed to determine if you might be a candidate for such a procedure.