Yes, it is possible to get red eyes when the bit of dandruff gets into your eyes because of the coming bacterium. The bacterium in the dandruff will stimulate your eyes. You should wash the eyes as soon as possible. Then you could do the warm compress which will help you get comfortable eyes soon. Your eyes will be moisture and comfortable.
Ok, I can see you are worrying about the possibility that dandruff would give rise to red eyes, but shall I remind you that dandruff actually does not affect your eyes, at least according to the vast majority of cases. Because red eyes is primarily resulted from bad use of your eyes or work stress or dry eyes, etc. So, just try to take it easy. Perhaps your anxiety would lead to red eyes. Just try to pay attention to your diet and get more exercises, and make proper use of your eyes.