Summarize the following:
Use your fingers to work the oily substance into the fur around the gum. Make sure the affected area is completely covered.  If your dog licks off some of the peanut butter, it's okay, as long as you use a peanut butter that doesn't contain xylitol.  Dogs can eat other oils, as well. However, try to keep your dog from licking at the oil, as they may get diarrhea or suffer from pancreatitis if they consume too much. This gives it time to soak into the gum, loosening it. Sit with your dog so you can make sure it doesn't snack on the oily substance or the gum. Keep the chewed gum out of the dog's mouth. Sugar-free gum is harmful to dogs because it contains xylitol. Although regular gum is usually harmless to dogs, it's best to play it safe. Insert the teeth of the comb down into the fur just above the gum. Then, pull the comb down through the fur to dislodge the gum. If the gum doesn't come out easily, apply more oil.  You can also pick the gum out with your fingers. If there is still gum caught on your dog's fur, you may need to apply more peanut butter, olive oil, butter, or mayonnaise. If 2 to 3 applications don't work, you may need to trim the dog's fur. Rub dog shampoo into the pup's fur, working it into the affected area. Then, rinse your dog thoroughly to remove any remaining soap. Dry your pup with a clean, dry towel. Use a degreasing shampoo, if possible. It will more easily remove the oil. As an alternative, wash the affected area with dish soap before you wash the dog.

Summary:
Apply peanut butter, olive oil, butter, or mayonnaise to the gum. Allow the oily substance to sit on the gum for at least 3 minutes. Comb through the dog's fur to remove the gum. Shampoo your dog to remove the oil and any remaining gum.