Some questions to consider: is this a new behavior? Is it both front feet or all feet? Are there any other conditions present such as limping or swelling in the foot? A foot licking problem can start out as a medical one and later become a habit or compulsive behavior.
Your veterinarian will be the only one to fully determine what is going on with your dog's feet, but here are some things your veterinarian will want to know and will be looking for on examination:
Are the feet red, swollen, or crusty/flaky?
This could be indicative of a local irritant (such as deicer) or inflammation/infection from bacterial, fungal and/or parasitic sources. Even if the inciting cause is no longer present, constant licking and chewing can become a self-propagating cycle of continued trauma to the skin and continued inflammation. (Also known as pyotraumatic dermatitis.)
Foreign bodies, such as grass awns, are painful and a common source of infection for the feet.
Arthritis or other painful internal conditions causing pain in the area without visible infection, etc. on the foot could also be a cause for licking.
Are there any irregular lumps or bumps deep between the toes or foot pads?
Cysts or other growths or small abscesses can occur, causing discomfort and licking.
My dog is just licking without any visible signs of something wrong on the foot!
Foot licking can be simply a habit behavior as well; seen when the dog is relaxing, stressed, or bored. Some dogs even chew at their nails with this type of behavior.
Depending on what your veterinarian finds on examination, treatment to stop this behavior will be aimed at the underlying cause. For cases of allergy or infection, there are medications and/or dietary changes that can be made to assist with the problem. Learn more about food allergies and atopy (inhalant allergies).
In situations where pain is the underlying cause, that should be dealt with directly to alleviate the licking. Growths or abscesses are usually treated surgically.
For difficult cases, a visit to a veterinary dermatologist or university veterinary teaching hospital may be in order.
Behavioral modification to stop paw licking and chewing, like any behavioral modification, takes time, patience and consistency. There are several topical products that can be used to discourage this behavior (bitter, hot taste, etc.). Physical restraint, such as an e-collar is often used for medical conditions to allow the foot or paw to heal and thus take away the inciting cause to lick.
Distraction is also a good technique: playing games, offering other toys and incentives to keep your dog occupied coupled with positive reinforcement will help break the cycle. If additional behavior help is needed, consider working with a specialist in veterinary behavior (Click on "Diplomates" to find a veterinary specialist in your area).