A black cat is not going to be getting burnt any time soon, but breeds with white or sparse fur are at risk. This can show up as red and warm skin tender to the touch. A cat who has been burnt in the past is at a greater risk of skin cancer.A black cat is not going to be getting burnt any time soon, but breeds with white or sparse fur are at risk. This can show up as red and warm skin tender to the touch. A cat who has been burnt in the past is at a greater risk of skin cancer.