I get a burning, freezing, wet sensation in my feet when I put shoes on? What might be the cause?

Peripheral Neuropath. Your description of burning in your feet with a sensory stimulus is typical for a small fiber peripheral polyneuropathy. You should see a neurologist to get a further workup and hopefully determine the cause which will help guide treatment. Good luck and best wishes.
Foot issue. Unless you have a rash on your feet that may be suggestive of a fungal infection, another reason could be neuropathy such as compression of a nerve in the foot. A good physical exam of your foot may help pinpoint the problem. Sometimes nerve conduction study may help shed light.