Because sweat contains salt and urea,which is also known as uric acid. After running, sweat get in your eyes, then it stimulate your palpebral conjunctiva, resulting in the burn eyes. I advise you to wear a headband or a headsweats skull cap next time you work out, and when sweat drips into your eyes, do not rub it with hands, just try to blink it away.
Because your body creates a lot of heat energy, you will feel hot when you run. However, there may be the eye inflammation for your eyes. You'd better go to check the eyes to avoid some eye disease because there is much bacterium during your running which may come into your eyes.