When I am sleep deprived, my eyes get super dry. Why is this?

Yes. Sleep disturbance can cause a decrease in tear production and dry eyes. Also, decrease in sleep can lead to just tired eyes without decrease in tear production. See an eye doctor to determine if you have an underlying eye condition such as dry eye syndrome, allergic conjunctivitis, meibobian gland disease, thyroid disease, to name a few.
Due to exposure. We all need the proper amount of sleep. However this is variable among people. When you sleep, your eyes are closed, thus allowing them to rest and regenerate. By having the eyelids closed, moisture from tears stays on the surface for longer periods of time.