If a renter is late with the rent payment, does the late fee get assessed the 15th or the 16th of the month?

This depends entirely on the tenant's lease with the landlord--the law does not address this issue.

If the lease does not provide for a late fee, the landlord may not charge one (he may seek to evict for late or nonpayment). If the lease does provide for a late fee, it will be assessed on the day/date set out in the lease.