Tears function in keeping the eyes lubricated and reduces friction when blinking. The act of blinking helps to keep the tears in the eyes and push them backwards. There is a connection between the eyes and the nose via the tear duct. The tear duct (also called Nasolacrimal Duct), drains out on the face and also runs down to drain into the nose.
When tears get into the nose through the tear duct, it mixes with mucus, making the mucus in the nose very liquid or dilute. This is why the nose "runs".