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So you are sucking a lot of cold dry air into your throat and lungs which drys out the natural mucuses that exists there, and causes temporary inflammation and pain. This results in that burning sensation you feel in your throat/chest/lungs, which may or may not be accompanied by wheezing and coughing. It can be quite painful and quite distressing, but usually, it should subside within, hmm, about 15-20 mins after you stop exercising.
A burning sensation is a particular type of pain distinct from dull, stabbing, or aching pains. A burning kind of pain is often related to nerves, but there are many other potential causes. Injuries, natural wear and tear, infections, and autoimmune disorders all have the potential to cause nerve damage and pain. Many causes of a burning sensation have no cure, but treatments are helpful in controlling the pain.