I heard using cold milk as an eyewash is good for your eyes. Is this true? Could it be harmful?

Don't do it. While I have never taken care of a patient who has done this, i cannot think of a single reason to think this would be a good idea. I would recommend using an eyewash or simply artificial tears.
No. Cold milk as a routine eye wash is neither good nor bad for the eyes. The only use for milk as an eyewash is if you get pepper spray or actual pepper food dust in your eyes. The fat in the milk will then help dissolve out the stinging particles.
No, but not helpful. Cold milk is only advisable if you do not have acess to sterile saline to irrigate your eyes. The lipids in milk may have a negligible effect on certain fat soluble sprays such as tear gas (cloroacetophenone), or pepper spray.
Not beneficial. Your own tears contain 30+ disease fighting chemicals. I usually don't recommend "washing" the eyes with anything foreign unless you get dust or a chemical in them. Sounds like a waste of good milk.
Use water. If your goal is to rinse something out of the eye, use plain water or over the counter eye wash. If your goal is to lubricate the eye, use over the counter artifical tears. Milk is not sterile and holds no benefits that I am aware of. The primary ingredient in your own tears is water, so i'd stick with that.
Could be-better not. What is your "purpose" of using eye drops? If its for common problems such as dry eye or eye allergies. There are drops that are specifically formulated for these problems. They are better choices. These come in different strengths depending upon the severity of your problems and your eye doctor can help you choose the best one for you. I don't know any study that shows milk to be beneficial.