It's more complicated than it seems: the functional elements of a 3D print can't be copyrighted, but they may be blended with decorative elements that can be; what's more, if we err on the side of caution by "open licensing" stuff that isn't even copyrighted, the effort to open up copyright ends up normalizing the application of copyright to new subjects.
Michael Weinberg's 3 Steps for Licensing Your 3D Printed Stuff [PDF] is the best read you'll find on the conundrum with copyright and 3DP, a guide to the contours of a law that has been distorted beyond recognition in an effort to fit it to something that was inconceivable at the time of the law's forumulation.
Weinberg penned several important, eminently accessible papers and posts on copyright and 3D printing for Public Knowledge: It Will Be Awesome if They Don't Screw it Up: 3D Printing, Intellectual Property, and the Fight Over the Next Great Disruptive Technology; Is it legal to print Settlers of Catan tiles on a 3D printer?; and What's the Deal with Copyright and 3D Printing?.
Copyright as it relates to a printed object's digital files is where 3D printing really starts
to feel different from a more traditional copyright analysis. Normally, we do not spend
very much time trying to make distinctions between a digital photograph and a file that
contains a digital photograph; in that case copyright can be thought of as protecting the
entire bundle, which means that the distinction is not particularly meaningful. In large
part, this is because the photograph itself is well within the scope of copyright protection.
However, the distinction between a digital file and the work that it represents can become
meaningful if the work in question is a non-copyrightable object. In at least some cases,
a digital file that represents a non-copyrightable object (such as, say, a screw) will be
protected by copyright even if the screw printed from that file cannot be. Notably, a
copyright for a digital file does not automatically give you the right to control the use and
reproduction of the printed object itself—something that will be discussed later in
"Files" on page 14.
3 Steps for Licensing Your 3D Printed Stuff [PDF] [Michael Weinberg/Public Knowledge]