It's almost impossible to damage your lungs by breathing cold air when you exercise. More than 70 percent of the energy produced by your muscles during exercise is lost as heat. This extra heat can be used in your nose and bronchial tubes to warm the air as it travels to your lungs. Air inhaled at 40 degrees below zero Fahrenheit will be warmed to more than 100 degrees by the time it reaches your lungs, so your lungs will not suffer from frostbite.
However, air that cold will burn your nasal membranes and cause pain in your nose that is so severe that you will lose interest in exercising very quickly and look for shelter. To protect your nose in very cold weather, you can cover it with a scarf, balaclava or face mask placed over your nose and mouth.