I can't really visualize the amount of momentum involved, or how sharp the chestnut is. Is that specifically about eye injuries, or could it hurt someone some other way?

If you whip it above your head at a good 100-200 rpm, anything is possible.

After writing the above comment, I watched videos of people playing conkers and now I understand how it could, in theory, cause an eye injury. It was hard for me to visualize how close the defender's conker is to the defender's body before seeing the video. I was somehow wrongly imagining that it was being supported on a much longer string or with the help of other objects somehow.

Literally the first video I watched, 2:28 min in, kid gets knocked in the teeth: https://www.youtube.com/watch?v=egBjZaKNLuc

Ah yes "accidentally"... ;-)