If your Zillow Rental Manager listing is no longer available for any reason, you can quickly expire the listing from your Listings page. This will stop it from syndicating to other sites and those sites will then take the listing down.
Only for For-Sale listings, if your listing is Pending, you can set this from the listing's Description page by checking the box for "Sale Pending" under the Property Website URL.
Zillow Rental Manager does not determine whether a listing is Sold. Zillow will update the property record based on third party data after the close and the listing is expired.