Post-closing literally means after closing. It's a catch all phrase for jobs or tasks that take place AFTER the actual closing. Post-closing includes the final takedown/bringdown of title and recordation of documents, the issuance of title policies, disbursement of funds not disbursed at the closing table, stuff like that.
Sometimes lenders and title agents use it in their day to day lingo like "Oh, we'll take care of that post-closing." That just means they'll do it later and won't postpone the closing for the sake of whatever it is they need to do.