The skin usually turns black and becomes very hard to the touch after it dies. It will fall off eventually if left in place but it is better to remove the dead skin to allow healing to occur faster and to ensure that the abscess can be properly treated. Abscesses that occur around the base of the tail are very prone to causing skin sloughing. Early aggressive treatment is best when there is pain or swelling around the base of the tail. Abscesses are generally treated by opening them up to allow them to be adequately cleansed with an antiseptic.
Abscesses can be very painful.
If a cat will not allow an abscess to be opened due to the pain it may be necessary to use an anesthetic agent so that the abscess can be properly treated. If there is dead skin over the abscess is should be removed as well. It surprises many clients that this can often be done without an anesthetic agent, since the dead skin has little or no feeling. The live edges still have nerve endings and many cats do object, though. After the abscess is cleaned most vets will prescribe antibiotics.