Dog chewing on feet

Pretty common for dogs to get yeast infections on their feet. Some dogs will lick or chew on their feet because of this. Yeast makes them itch. Best way to treat yeast infections is with their diet. Certainly don't be giving them starches or carbs as this will inturn help feed the yeast outbreak.
Are the dogs nail beds red or inflamed a little? Does its feet smell like corn chips? If so bathe the dog and gently clean it's feet. I'd use a soft tooth brush and gently scrub the nail beds. Bathes need to be regular for a little while to help rid the dog of the infection and rid it of the dead yeast. After cleaning dog, mix some vinegar and water 50/50 and spray the dogs feet and nail beds. This will help kill the yeast but the dead yeast does need to be removed or will flare up again probably. If it has yeast infections in the ear or any where else or is suffering from it all over its body such as reddish skin, just addressing the feet probably won't do you much good.
Broad spectrum antibiotics allow yeast blooms because most antibiotics wipe out good bacteria that keep the yeast levels in check. All dogs carry a little yeast in their systems and are subseptible to yeast outbreak as a result of going on antibiotics for treating infections or infection prevention. Most with naturally healthy immune systems never have a problem with starches and carbs but some dogs do and when a dogs immune system can't take the stresses of carbs and starches in kibble after a few years, just the food itself will cause these dogs to have chronic food allergies just seemingly out of the blue. Doesn't necessarily have to be from antibiotics causing an un natural balance of yeast in dogs.
Hopefully you have discovered what's wrong with your dog by now but if not, you might consider something causing it along these lines