For Origami from unusual materials check out the work by Polly Verity (Plastic) and Ilan Garibi (Metal).
Ulrike Krallmann-Wenzel folded some stuff from pure silver.
Eric Gjerde is experimenting to obtain paper like stuff by growing microorganisms and pressing them into thin sheets.
Napkins and cloths have been folded by very many people.
Don't know if this counts too but some artists like for Example Sipho Mabona use casts to produce a big quantity of models in a short time.
Robert J. Lang made some huge sculptures (and solar sails) based on his Origami models, but I'm not sure if this is what you are looking for.
There are also Origami folding sheets available to buy that are not paper, beginning with all the aluminium laminated stuff, plastic up to metal mesh. I think I also heard of someone using very thin sheets of wood that he soaked in water to make them pliable.
Amongst the weirdest stuffs I folded was Won Ton dough
https://www.flickr.com/photos/origami_8 ... 3539060127
Not sure whether I ever tried to fold from plasticine, but I've definitely seen videos of someone folding with thin sheets of Fimo, that she first put a few times through a noodle machine to get paper thin sheets.