Natural model: Tree leaves
photo credit Growing leaves takes a lot of energy from the tree, but it provides the tree with the needed sun energy by means of photosynthesis. They also capture sequestered carbon dioxide and convert it into oxygen . During the winter months, when sunshine hours are reduced, the tree preserves its energy by dropping the leaves onto the ground. There they perform two functions. First, they protect the root system from cold temperatures. Secondly, they decomopost and become nutrients to the fungal cycle which in turn converts into soil to help the tree grow. In this closed-loop system, there is no waste. Energy and matter gets recycled into a system that can benefit from each stage.
Once they fall to the ground they provide a protection layer to the environment around the tree, through decomposition, they add nutrition to the mycorrhizal layer (fungi and root system) and become part of bacteria and soil in order to foster new growth.
Can your design be recycled? How toxic are the components? Even when a design is printed on paper, there are many factors that can make it hard to be recycled. Paper production itself can add chlorine compounds and other chemicals to the paper fiber, and the ink the message was printed with or the finishing process like aqueous coating can contain toxic compounds. For information on solutions to good paper/ink vs bad paper/ink read Sustainable Graphic Design by Wendy Jedlická et al.
Paper with enzymes
This is not a realized idea yet, but while researching fungi and bacteria effects on substrates I had this thought that perhaps enzymes could be infused into papers that, after a certain trigger, would convert any toxic particles into benign compounds. Then, once the piece is held for a certain amount of time, bacteria would be activated that would start the decomposition of the paper. This would allow a design to be thrown onto the compost pile, returning resources to the earth.
This idea would have to be explored by chemists as well as biologists to see if it is feasible.