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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.
There are quite a few different medical conditions that can cause a throat-burning sensation. In most cases, a truly intense burning feeling in the tissues of the throat suggests that the issue stems from the stomach. There are, however, many non-gastrointestinal issues that can cause a slight burning feeling in the sensitive tissues of the throat and neck.