Many Class C airspaces do not have a full-time tower and the airspace reverts to Class E or G when the tower is closed. The wording in CFR Part 91 § 91.225 doesn’t specifically say whether ADSB is required when the tower is closed and I can’t find any guidance from the FAA or AOPA on the matter.
(d)…no person may operate an aircraft in the following airspace unless the aircraft has equipment installed that meets the requirements in paragraph (b) of this section:
(1) Class B and Class C airspace areas
(3) Above the ceiling and within the lateral boundaries of a Class B or Class C airspace area designated for an airport upward to 10,000 feet MSL;