Perhaps is cause irritated eyes if you get hair dye in the eyes. If you didn't hair dye into your eyes, it shall not irritate your eyes. As for the itchy eyes after dyed hair, it can be caused by many other problems such as dry eyes, and tired eyes etc. Also, you shall check if your eyes are burning or red. If so, it is probably caused by eye infection. Just consult your eye doctor.
There are some people got irritated eyes because hair dye. So, it can be one of the causes for your itchy eyes. But usually, people get irritated eyes by contact the hair dye. If you didn't contact it, it should be other factors that contribute to the itchy eyes. And the most common causes is dry eyes. If so, applying some eye drops that can help you. good luck.