Answer: It would affect the lacrimal apparatus, the system in charge of the production of tears and their drainage system.
Step-by-step explanation:
If a blockage where to happen at the lacrimal punctum, or if it is narrowed, the tears would not be able follow their normal path, which is to move into the lacrimal sac, to be drained into the nasolacrimal duct afterwards, so the tears would overflow from the eye onto the face.