Is there any way to use Preview to view files on another server? I own a Mac and ssh into a Linux server.  How can I view a file on the Linux server using the Preview program on my Mac, without having to explicitly copy it to my Mac beforehand?
You cannot as Preview runs on the Mac and loads all the data into itself before rendering it.

In general this is how all OS X programs work, the data has to be transferred into the app running in the Macs memeory

What can be done is using X11 the normal Linux display mechanism, which was designed to work across a network. You can have X11 programs running on the server that display on another machine. Unfortunately Apple's GUI does not support this (an earlier version did but we suspect for the cost of licensing this was replaced)