From what I've seen, not necessarily. I've seen more than a few job posts with multiple hires still showing up in search results.
I think the only time they get deleted automatically is when there's been no activity on them for 30 days, then they expire. But for a concrete answer, I'd wait for a mod or staff member to reply.
Once you hire a freelancer from their proposal, you'll be prompted to select if you'd like to close your job post or keep it open. Job posts are automatically closed after 30 days of inactivity.
Let us know if you have any questions and I'd advise you to review our client resources for more information about using Upwork as a client.