I was faced recently with wanting to move Asana projects from one workspace to another. Unfortunately this isn't a feature in the Asana interface (yet).
There's a PHP script floating around GitHub which will copy tasks between projects. This requires you to do some manual checking of project IDs, and you need to create the target project in advance, so I've extended the script and written a front-end web interface for it.
It's very, very rough, but it does work.