The word "post" means after. So any reference to post closing means after closing.
We have a post closing department which handles all of the details and work that takes place AFTER the closing. For a title insurance agent, that means recording documents, issuing policies, delivery of documents, filing, scanning, escrows and sending out checks, etc.
Mortgage lenders also have post closing departments which process the mortgages for sale in the secondary market and/or for loan servicing.
If you are a consumer and someone asks you to do something, perhaps send a document and they say, "You can send it post closing." they just mean it's okay to send it to them later, after closing.