I am sure that birds don;t have tear ducts like human eyes.
They do have a semi-transparent membrane which crosses the eyes much in the same way that we blink, and this no doubt protects the eyes of a flying bird.
If ever you have the opportunity to look at a swift close up, you'll see that they actually have eyelashes in front of their eyes. Maybe they are there to help stop eye-watering in these fast flyers.