How do birds deal with their eyes drying out at high speeds?
Birds have a body part known as the nicitating membrane otherwise known as the "third eyelid". This part has become vestigial in humans, where it remains as the plica semilunaris. 



This image of a masked lapwing clearly shows its nicitating membrane in action, where it covers the eye in a horizontal motion. This is analogous to blinking in humans, and the membrane moisturises the eye and removes debris while the bird is in flight. 



This paper explains in detail how the peregrine falcon (the world's fastest bird) clears and moisturises its eyes when diving at speeds of over 300km/h.