I don't believe that you will cause permanent damage to your lungs when running in the cold but yes you can cause irritation to your lungs when you're breathing in frigid cold air too much.
Frigid weather can also make it harder to breath and take your breath away so if you want to job or run in the cold weather it's fine to do so but don't stay out too long or over exert yourself and take frequent breaks while running in the cold weather.
Also wear a scarf over your mouth to keep the air you're breathing in as warm as possible to avoid irritating your lungs.