There is a feature in Moodle forums called "Export whole discussion to portfolio".
However, there doesn't seem to be a feature similar feature in advanced search where you are able to search for the posts you want to display and export only the posts you want to see.

An excellent example would be if an instructor wants to see how many posts or only the posts from a certain student. And wants to save a copy of it. The instructor can use the "search forums" search to drill down the criteria but does not have an export option like the "Export whole discussion to portfolio" option.

It would be great to see a feature like this in the future.