Consider the following situation: you're preparing dinner, and handling hot peppers. You start doing something else, and, forgetting about the peppers, rub your eye(s).... What do you do?
Capsaicin, the source of the horrible, horrible burning pain, is a non-polar molecule. This means that it's not soluble in water, unless you add something like soap, which has both a polar group and non-polar groups, allowing it to interact with everybody. Because it's not soluble in water, flushing your eyes with water will mostly spread the capsaicin around, and as anyone who has accidentally gotten shampoo in their eyes can tell you, soap isn't that much less painful (though it does allow you to eventually flush it out with water). This is why milk is more effective for calming a burning mouth than waterSo what other options are there?
If you, or someone you live with happens to wear eye makeup with any sort of regularity, there is very likely makeup remover somewhere in your abode. It's explicitly formulated for removing oily things like mascara from the sensitive eye area without causing irritation. This is also why eye makeup remover is a remarkably effective capsaicin remover.
So next time this happens to you, reach for the makeup remover.* If you regularly handle peppers, even if you don't wear makeup, it's good stuff to have around.
*Mr. ME will totally vouch for this