The dead parts of the comb will just shrivel up and fall off, but you do run the risk of infection. You may consider just going ahead and dubbing it.
Whenever mine get hurt I wash the wounds with sterile saline (1 qt boiled water and 1/4t salt) and generous amounts of Neosporin. They may act depressed for a day or so, but they'll bounce back quicker than you think.
Personally, I don't give antibiotics or pain meds as a general rule and mine have healed perfectly without. I had a pullet that was laid open from the top of her head to she shoulder, about 6" long, 3" wide with muscle and bone showing and she healed perfectly with saline and neosporin.