Anyone who has had any burns at all recognize that there are big burns and there are small burns but all burns are potentially serious. Even a small burn on a finger, as you described, can be rather distressing it has the potential to result in loss of range of motion of the affected finger as well as a decrease in hand function.
Once a burn wound is closed it still has a rather long period of common of healing involved. The entire process may take 12 to 18 months. Once the burn wound has closed it goes through a period of remodeling, and this is the period in which scar formation will develop. It is the development of scar particularly across joints that can affect range of motion and function.
A scar that you describe on a finger though not very large can have significant impact on the function of the finger in hand. Typically when treating a burn of this nature a patient is seen regularly by an occupational therapist who is focusing on the preservation the range of the affected finger. This is accomplished by teaching the patient a series of exercises which should be done several times a day. Also the occupational therapist will typically measure the hand for some tightfitting pressure gloves that will control the development of the scar.
Certainly my advice to you is to no longer try to excise or revise the scar yourself. If this burn scar is causing you this much distress, I would recommend that you see someone who has experience in the treatment of burn injuries. The first options should not be surgical, but referral to a hand therapist with experience in the treatment of burn injuries.