How can dying your hair affect your eyes?

Irritate. The what of the blow dryer could have an irritating effect on the skin around the eyes both from the heat and hot air. If this is concern be careful how you use blow dryer, facing it away from the face. If there symptoms that do not respond to conservative compresses and eye drops you should see eye doctor for reassurance.
Drying hair eye pain. One can burn the cornea of the eye or dry it excessively from the hot dry air of a blow dyer when drying the hair. Be careful and don't aim the dryer too close to your eyes.
Irritation or rash. Hair dyes contain chemicals that some individuals may be allergic to called contact dermatitis. You can get a rash around the sides of the face and neck and eyes due to contact to the chemical in the hair. At the very least even if not allergic, you should avoid getting the hair dye in your eyes because it can cause burning and irritation.