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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.
Allergies, especially those due to airborne particles, cause irritation and a burning sensation in the chest and throat. Pollutants or certain foods can irritate the throat and also cause inflammation, itching, nausea, and headache. An inflammation of the esophagus is known as esophagitis, causing a burning sensation in the throat. The condition also causes discomfort, chest pain, and excessive belching.