Blocked tear duct occurs when the duct that normally allows tears to drain from the eyes is obstructed or fails to open properly. It often happens to newborns. And adults get blocked tear duct because of aging, a thickening of the tear duct lining, nasal or sinus problems, infections, abnormal growths such as tumors or an injury to the bone and tissues around the eyes. So according to experts' idea, I think that only cold will not cause tear duct unless cold cause some nose problems.
But if you get blocked tear duct, the type of treatment for a blocked tear duct depends on the cause. If it is caused by infection, you may get treated with antibiotics. But if it's caused by a problem with the size or shape of tear duct, you may need surgery.