Permit me to supplement Andrey's definitive answer.
First, as Gerry Myerson says, this problem is discussed in Robert Lang's Origami Design Secrets: pp.315-318, under the title "The Margulis Napkin Problem." He credits the problem to Gregori Margulis.
Second, the problem is discussed in Igok Pak's book Lectures on Discrete and Polyhedral Geometry, p.354, which is available online. You can pretty much guess the proof from the following instructive figure of Igor's:
Napkin http://cs.smith.edu/%7Eorourke/MathOverflow/NapkinFolding.jpg
Third, there is another surprising result that is intellectually analogous to increasing the perimeter by folding: The volume enclosed by any convex polyhedron can be increased by bending the surface (retaining intrinsic isometry) to render it nonconvex! See Chapter 39 of Igor's book, p.339ff.