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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.
If your chest burns when coughing, you may be suffering from acid reflux. This is when the stomach acid seeps up into the esophagus from the stomach. Acid reflux can cause a cough, often at night accompanied by pain and burning. This is when the cartilage that cushions the ribs becomes inflamed.