Hogwarts is closed during July. The only member of staff who stays on in an official capacity is the Caretaker, Mr. Filch (SN). However, school-related work does get done, such as sending school letters inviting the incoming First Years. O.W.L results are mailed out by Owl Post, although these would be sent by the Ministry, not the school. Presumably, the Head also spends time in the summer months setting staff assignments for the upcoming year.
Commentary
Notes
Rowling's statement that no staff stays at the castle in the summer, from the South West News Service interview in 2000 (SN), probably doesn't take into account some of the character development that happened in subsequent books and other writings. For example, since McGonagall does live at the school (WW), it is likely that she stays on campus through the summer. The same might be said about Trelawney, who seems rather solidly entrenched in her tower, but we don't know. The spirit of Rowling's statement suggests that staff members don't work at Hogwarts during the summer. However, since summer work is being done, notably the letters sent to First Years, it is clear that McGonagall and Dumbledore at least do some work during July and August.