A Google image-search for "folding guitar" yields many examples; obviously none of them set neck. For some reason, I have a vague recollection that either Hoyer or Shergold used to make them, but I might have an erroneous memory there. Certainly I'm not seeing any in the image-search.
In addition to the string-tangle, I'd be concerned with stress on the neck; particularly for acoustics. Relief on the neck is predicated on back-tension provided by the strings. When that back tension is removed by folding the neck over, the neck can bend in the wrong direction. I've likely mentioned in past that when I brought my pre-war Kalamazoo to Brian Dubbledam to address the buzzing I was getting, he asked me what gauge strings I was using. When I answered "light gauge", he identified that as the problem and urged me to switch back to medium gauge, and the problem would go away. I did, and it did. The non-adjustable neck had a degree of relief built in to result in a straight and level fretboard when the pull of the strings in one direction and the bow of the neck in the other, were in equal balance. Too little pull from the strings, when using light gauge, and the neck bow won.
That doesn't mean a foldable neck would necessarily result in instant neck damage. But certainly leaving the instrument folded up for any lengthy period would allow the neck to conform more to the relief during that period. And forcing the neck to go back and forth from high tension to none can't be good for it. Of course, if we're talking about a heavy maple reinforced double truss-rod neck, and regular use of 9-46 sets, such concerns are moot.