Why would a foal's eyes run constantly?

There are several reasons why a foal might have tears running from its eyes constantly - including an infection or allergies (Yes, horses can have runny eyes from allergies, including to pollen and, worst of all, hay).

Another possibility is "atresia of the nasal puncta" which is another way of saying the tear duct between the eye and the nose is blocked. (I had a dog with this). It's uncommon in horses, but far from unknown. These days it can be fixed with simple surgery - the vet will apply anesthetic and then make an appropriate hole.