A board of county commissioners operating a transit system or a county transit board created under sections 306.01 to 306.13 of the Revised Code shall be exempt from and shall not be required to pay any taxes on property, both real and personal, belonging to the board, which is used exclusively for any public purpose; provided, such exemption shall not apply to any property belonging to the board while a private enterprise is a lessee of such property under written lease providing tenancy for longer than one year.