It seems that if something can be stretched then it can be folded along a curved line. Since paper can't be stretched I can't fold it along a curve. But it's just an observation not an answer to the question. Does anyone know the answer? Is it because of some kind of area conservation?
Apparently, while you can't fold paper along a curve in 2 dimensions, you can do it in 3 dimensions.
This article from the journal Physics describes how a flat ring of paper, folded along its circular center, deforms into a saddle as the material buckles to absorb the stresses.
(Longer discussion at Physicsworld courtesy of an answer to a related question.)
The short path between 2 points on the paper before folding is a straight line, all points of which belong to the paper.
Now suppose that we managed to completely fold it along a curve between these 2 points. All paths between them on the paper are now curves, with bigger lengths compared to the straight line, (which passes out of the paper).
So, it is not possible to fold it along curves keeping the same distance between all points. And it requires streching to change distances.