Dogs dig instinctively but there may also be other reasons. Hot temperatures outdoors may cause a dog to dig a hole in which to cool himself. Provide plenty of water and a cool place for him. Also, lack of exercise or boredom might cause him to dig so be sure to provide plenty of playtime and exercise. Your dog could be burying his "treasure" or, chasing a scent. To train you dog to stop digging, state the command no when you catch him. Then distract him with play or exercise.
These may be ways for dogs to release pent-up energy, so try to make sure your dog or puppy is receiving plenty of exercise. Do not forget to interact with him. He is a pack animal and he needs you to be social with him. Dogs may dig holes in the summer outdoors to lie in so they can keep cool, a dog may dig a hole in search of a mole or rodent (terriers were bred specifically for this task), and remember the largest backyard in the world with fifty dog toys in it will still become boring after a while. There is no substitute for human interaction.
Make sure your dog or puppy is getting plenty of exercise. Teach him some new commands or tricks so his brain will be stimulated as well. Before you correct him for chewing on your sneakers, make sure you have not confused him by giving him an old sneaker to chew on. Use only toys which are made specifically for chewing. As a general rule of thumb, if he can get the whole toy in his mouth he may be able to swallow it whole or it could become lodged in his throat. When you take an object from your dog or puppy that you do not wish for him to have, supply him with a chew toy in which he can have, and offer praise to him so he can associate what things he can chew on and which things he cannot - be consistent. Offer ice cubes to teething puppies - the ice will help relieve some of the discomfort of his gums and aid in helping to cut teeth. (Make sure you run him outdoors occasionally to eliminate - ice is water).
Chewing will occur when a puppy is teething, which could be for as long as one year old, or stress can cause a dog to chew (just like children or adults bite their nails). Dogs need to chew to relieve stress and excess energy. In fact, puppy teething is a natural part of development. If your dog chews on something that is off-limits, say "no" or "eh" in a low voice. Then give your dog a safe chew toy. Praise him for good behavior. Remember to give your dog safe chew toys instead of old household items like old shoes, because that could encourage him to also chew on new shoes. You will also want to avoid rough play and games like "tug of war" as this will encourage aggressive behavior.