Can OS X access files in my Dropbox account without using space on my local hard drive?
Update: See nassimhddd's answer below about ExpanDrive allowing you to mount your DropBox account as (external) drive - since
  its Mac version 3.0.3021 May 2013.


The short answer to this is: No. As far as I'm aware, you can't mount your Dropbox account as some sort of remote file share - its synced files are always written to your local disk.

That said, you can change the location of the Dropbox folder away from its default of ~/Dropbox, so if you have a separate internal or external disk you could put the Dropbox folder there. 

You may encounter problems if Dropbox occasionally can't access the detached volume holding its folder, however. See here:


  Your Dropbox folder must be available when your computer boots and
  before the Dropbox desktop application starts. Some USB drives, for
  example, take some time to mount before your computer recognizes them.
  The Dropbox desktop application could start before such a drive is
  ready and get confused. You will receive an error and Dropbox will
  give you the opportunity to relink your Dropbox account or exit if it
  is unable to find your Dropbox folder.