It really depends. Deer will eat most any plant if they are hungry enough, but American persimmons are not high on the preference list. I have them growing natively and I've not seen them browsed. I have high deer densities. Last fall I planted Nikita's Gift which are a hybrid. Not knowing how they would react to the cross, I tubed most of them. However, I had a few with branching and I wanted to collect one of the branches for scions. So, I left about 1/2 dozen of them completely unprotected. I happen to have a remote sensing camera on the field with some of these trees. I get regular pictures of deer in the field.
Last Friday, I finished collecting persimmon scions. I found 1 small bit out of one of the NG persimmon branches. Could have been curiosity. The rest were untouched.
Now, I'm referring to browsing on leaves and branches. As for persimmons, deer love all of them. Kaki persimmons don't fall from the tree. They can be astringent or non-astringent. Deer will eat Kaki persimmons if they can reach them and they will damage small unprotected trees trying to get to them. They will wait for astringent persimmon to ripen but they will eat non-astringent anytime. All American persimmons are astringent and they fall from the tree. Deer eat these off the ground after they are fully ripe and sweeten up, so the trees generally don't get damaged.
Deer are not the only wildlife that love persimmon. Raccoon, coyote, bear, and many other mammals love persimmons.