No they shouldn't. Salaries would not be saved. Utilites wouldn't be saved because the light and heat would still be on the same number of hours. It might save on lunches but then some bozo would demand a free afternoon snack so we're back up to the normal food costs. It would save on bus maintainance and gas but that wouldn't offset our tax dollars enough. Then you'd have parents crying about having to hire babysitters for that extra day so no way would we ever go to 4 days in the public schools.
If they really wanted to save tax dollars, they'd make year round school with semesters starting and ending alternately so one group of students attends winter/spring/summer, another spring/summer/fall and the third group summer/fall/winter. That way cities could handle 33% more students so there's no need to build more campuses thus saving the taxpayers.
The year-round idea you propose has some merit. I still think the transporation and utilities costs would be lower with 4 days of school rather than 5.
There is no reason thermostat setting throughout a building should be the same when people are not present in those portions of the building.
And, of course, all maintenance, driver, and cafeteria help should be hourly (or hourly based) and not salaried.