Are DropBox files viewed on iOS devices taking up space on them?
Yes.  The files are cloud based until you favorite them.
Straight from DropBox:

How much can I store on my phone or tablet?
The Dropbox iPhone app can save as many files as your iPhone or iPod
Touch can hold or Dropbox space allows. If you find yourself running
out of space on your iPhone, removing files from your Favorites list
will clear up space on your device. If you're running out of Dropbox
space, you can delete files from your Dropbox, or you might want to
consider upgrading to a bigger Dropbox.