When I was doing a lot of hiring we wouldn't take the job posting down until we were done hiring people with that title.
It made a couple people furious because they assumed we were going to take the job posting down when we hired someone and then re-post a new listing for the next person.
One guy was even stalking LinkedIn to try to identify who was hired, without realizing that many engineers don't update their LinkedIn. Got some angry e-mails. There are some scary applicants out there.
Some times a specific job opening needs to stay open for a long time to hire the right person, though. I can recall some specific job listings we had open for years because none of the people we interviewed really had the specific experience we needed (though many falsely claimed it in their applications, right until we began asking questions)