It is really not a good thing. Dry ice is the solid form of carbon dioxide. It is used primarily as a cooling agent. Prolonged exposure to dry ice can cause severe skin damage and the fog can lead to eye problems. If dry ice enters your eyes by accident, flush them out with cold water, and clean water up with cold washcloth. Do not take any medicine self-care before you talk with your doctor. Dry ice really does harm to eyes. After handling your condition a few minutes later, see a doctor as soon as possible.
Dry ice is solid carbon dioxide, its temperature is much lower than water. When touching water, it almost equals to heat the dry ice, which will heat of sublimation and also reduce the temperature of the water, sometimes, even frozen. So if a bit of dry ice into your eyes, you should never expect to use lots of water to clean them. You can use a dry cloth to wipe it. Of cause you should see a doctor if the situation is serious.
Ok, looks like there is something wrong splashed into your eyes. From what I know, dry ice would not be of great harm to your eye health. Because that is mainly comprised of carbon dioxide, and you could simply flush your eyes with some water and get eye drops to treat the irritative feeling. Take it easy and be careful from this point on.