There are plenty airports that close, particularly those in urban areas. My regular airport, LCY, is even closed on Saturday afternoons (it's primarily a business airport and has not traffic at that time).

For some, it may be a simple question of demand, so you could have late night cargo flights. For others, it could be due to noise restrictions for the benefit residents in the flight path. Conversely, there are plenty 24-hour airports out there, too. LGW is an example, with the bleary-eyed charter pax checking in at some ungodly hour...