Hi Dandylion,
Welcome to the Mattress Forum! 
Your question is a very difficult question to answer because there are many different definitions of green, biodiverse or environmentally friendly. In addition to this … much of the information on the web is as much “greenwashing” as it is factual.
Most of the many questions involved in “how green is green” are almost impossible to answer because you would have to evaluate each step of the production process from the beginning to end (cradle to grave) and somehow “measure” or assign a “value” to every step in the process in terms of harm done to the environment, how renewable and sustainable the raw materials are, how they break down, whether they can be re-cycled and what is involved in the recycling, the energy and resources used in every step of production, the energy used in transportation, and many other factors … many of which are impossible to accurately quantify.
I think though that it would be “safe” to say that natural latex, natural and/or organic fibers, and innersprings (recyclable) would probably be the most “green” or “natural” of all the different mattress materials. Recycled PET would also be a candidate (made from a mix of recycled food grade plastic and virgin plastic) although it can only be recycled once and then you are left with the same problem that it needs to go into a landfill and doesn’t break down and it can still be energy intensive to make.
Polyfoam and memory foam don’t break down in landfills nearly as well as natural latex or other natural materials either (they just break down into smaller particles). There is an interesting article here on polyfoam including the new “plant based” versions which introduce a whole new level of issues into the equation including the effect of GMO soy and the destruction of the rainforest that is a direct or indirect result of soy production. There’s another short article here from Latexco (which produces latex cores) which indicates that 100% natural latex has a smaller carbon footprint than other materials. Having said that … there are also some who are concerned about the effect of natural latex production on biodiversity (as more latex plantations are planted).
In terms of Talalay vs Dunlop (assuming that both are 100% natural) … Talalay uses carbon dioxide as a gelling agent and Dunlop uses sodium silicofluoride and both use various, curing agents, accelerators, antioxidants, mold release agents, and other additives depending on who is making the foam (all of which is generally proprietary information and a closely guarded “secret”). If green was my only criteria though … then organic Dunlop latex would probably be my choice in spite of the premium price because of it’s higher latex content (compared to Talalay) and even though it’s questionable whether it is really much different from 100% natural Dunlop that isn’t certified as organic. I have seen no evidence that 100% natural Talalay uses more or less chemicals than 100% natural Dunlop except for claims on the web that don’t seem to have any facts (or comparative testing) behind them and are often based on what someone is selling.
The FTC also has a green guide summary here and their more detailed green guides here that outline requirements for “green” claims or marketing.
There is also more information about recycling mattress materials in post #1 here (with thanks to Sleep On Latex) and in post #2 here that each person can also use to help make their mattresses a little less harmful to the environment and keep them out of landfills.
Outside of the certifications mentioned in post #2 here … Oeko-Tex also has some additional certifications here that also certify for environmental and social issues and FSC also has a certification that includes environmental and social issues and the Rainforest Alliance also includes biodiversity and social issues and Eurolatex Eco Standard also includes a certification for harmful substances and VOCs and TUVRheinland also has a certification for harmful substances and VOC’s and there are a few others as well but these are much less common in the mattress industry. A PDF of the Global Organic Latex Standard 3.0 standard is available for download here and does address some social and environmental issues.
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Finding out the “truth” of all of this and going to levels of fine detail about every step of the process is like going down a rabbit hole where there are really no black and white answers and ultimately each person would need to decide based on their best judgement and their own personal opinion of what they consider to be “planet friendly”.
Phoenix