Is it possible to rename an open Word document without closing it?

Word opens documents in exclusive mode AFAIK, so this locks the files and prevents access by other programs until the lock is released by closing the files or Word itself. I don't think providing shared access to documents is a workable solution, otherwise it would surely have been implemented long back if it made sense.

Yes, it's possible!

At least on a Mac (OS X 10.8.5). Just Cmd + click on the file name on top of your opened Office document, that you want to rename. You then see the path where the file is located. Next you click on the name of the folder directly under the file name. The name then appears in a Finder screen, where you can adjust its name to whatever you want.

So no need to first close the file, nor to use 'save as' and removing the first file from the finder! (I don't know if the same or similar trick works out in Windows.)