Allergies and anxiety related behavior issues are the two most common reasons why dogs chew on their feet. Allergies need to be ruled out before behavior therapy is initiated. Allergy treatment is about control not cure. Treatment may involve frequent bathing, allergy diet, anti-histamines, omega-3 fatty acids, treating secondary infections, allergy injections, steroids, and immune modulating medication. Have your dog examined by a veterinarian to have the feet examine and start therapy to relieve your dog's discomfort.