If you wear eye makeup, even if it isn't waterproof, you still need to use an eye makeup remover. It is best to use a cotton ball or square with the eye makeup remover rather than a tissue. There is nothing worse to me than seeing someone with day old black eyeliner and mascara under their eyes. Not to mention that the eye area is more sensitive than the rest of your face due to the skin being thinner. So, using an eye makeup remover ensures you will not have any remnants of yesterday's makeup and it allows you to remove makeup easily without extra wear and tear on your eye area. Practically every makeup line has an eye makeup remover, and I have tried A LOT of them. When I was first gathering products for my makeup kit I spent probably an hour at Sephora with my hands covered in waterproof mascaras and liners to test multiple eye makeup removers. The $20+ eye makeup remover I found that day does not compare to my favorite for the past few years, Neutrogena's Oil Free Eye Makeup Remover.
If you aren't a fan of liquid eye makeup removers, or need a product that is easy to travel with try Almay's Oil Free Eye Makeup Remover Pads.
You can find both of these products at your local drug store or one of my favorite stores, Ulta.