In New York State, can a landlord charge a non refundable pet deposit for rentals?

If its in the lease, yes.

Actually... this gets interesting. It really depends on what the state law says. For example, California says that all deposits are refundable, no matter what they are called (security, pet, etc.). And the law trumps a lease, so if the lease says it's non-refundable, you can demand the money back when you leave.

Which is probably why some landlords do pet rent instead/as well. That's not a deposit, and they get it every month.