Weepy eyes, or epiphora, can be temporary (because of an allergy or irritant) or permanent (because of a structural defect). Some dog breeds are more prone to epiphora than others, such as the poodle, Maltese and Shih Tzu.
The tear ducts of some large-eyed dogs, Chihuahuas in particular, weep frequently to keep the surface of the eye moist and free of debris.
Weepy eyes can result from allergies to commercial dog foods. HolisticPetInfo suggests reading labels and avoiding "BHT, BHA or ethoxyquin and food colouring."
When eyelashes grow inward toward the eye or when foreign matter (like a speck of dust) gets trapped beneath the eyelid, a dog's eye weeps in an attempt to dislodge the irritant.
Dogs' tear ducts drain into their nasal cavities in part to keep the nasal passages and snout moist. Dogs with narrow or partially-blocked pathways may experience excessive external tearing, or weepy eyes.
VetInfo says the "rust" stains on fur from weepy eyes are because of iron-carrying molecules being broken down by bacteria. Though tetracycline can help, it's best to check for problems like glaucoma or uveitis.