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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.
Pulmonary embolism is the most feared cause of chest pain when breathing. Pulmonary embolism is caused by blood clots that go to the lungs mostly from veins in the leg and get lodged in the blood vessel of the lungs. They can produce inflammation in the lungs that can irritate the nerves in the pleural membrane.