if not recycled --> Transporting woodchuck to wood. Chucking wood using fuel, transporting wood to milling plant. Transporting extracted cellulose to paper plant. Packaging plates. Transporting plates to super market, transporting plates to your home.
Electricity from renewables + bio-degradable detergent + water + some fuel to pump the water to your place
Just the amount of mass that needs to be moved in order to get those plates from the trees to your place is way higher than the amount of water used by the dishwasher.
I'm not typically an answer-my-own-question kind of person, but this has been on the back of my mind for days. The challenge in answering this is that we do not have an apples-to-apples kind of comparison.
On the one hand we have dishwashers whose primary impact is energy they consume for electric motors / heating of water and the impact of creating the dishwasher (which will not last forever).
On the other had we have paper plates that use wood pulp and the energy requirements of transportation and production.
To make this an apples-to-apples comparison, we can compare based on cost. The heuristic is that when we pay for dishwashing or paper plates, we are paying for the summation of the resources that went into their production. Thus we can use cost as a proxy for environmental impact.
This website estimates that it costs $0.63 per load to run the dishwasher (cost of the dishwasher being factored in).
The cheapest paper plates I could find online are $18.86 for 600. If we assume 100% markup for retail items (i.e. the difference in cost it took the manufacturer to produce the product from the price they are charging in stores) then the cost per plate is about $.016 per plate.
Assuming that a dishwasher can wash 50 plates in a load, the equivalent for paper plates would cost $0.78.
Since $0.78 > $0.63 we can assume (under this hypothesis) that paper plates have a greater environmental cost than a load of dishes.
Note: The above estimates are based on some pretty big guesses. E.g. 100% markup for retail items. If there is %200 markup on paper plates then that reverses the conclusion. Further, as others have pointed out, even if we can accurately determine total $ cost, that does not not necessarly translate to environmental cost. Some part of production may have low $ cost, but high environmental cost.
If you only focus on daily running costs, then an efficient dishwasher that consumes only a few litres of rainwater, and electricity from solar panels on your roof, will win — hands-down.
However, that dishwasher may only last 10 years, and may only complete 3,650-ish wash cycles, before failing in a manner that requires its complete replacement. So if you expand your time-frame to 10 years then it is almost certain that mass-produced, "non-dyed, 100% natural, biodegradable" paper plates will have a lower environmental impact. You don't even need to do math to know that more resources will go into a high-tech dishwasher than 3,650-ish low-impact place settings.
We use plates and utensils made of waste sugarcane pulp when camping... so there's no need to even fell any trees... or to use 'paper' at all. The pulp is a waste product that would have otherwise been burned.
If you can latch onto a local supplier of low-impact disposable place settings then I think you're onto a winner.
Embodied vs working energy. You would have to calculate the embodied energy that went into the production of both products. At some point the dishwasher will seem like the winner here as it should last a good portion of the time and be repaired for a fraction of the embodied energy of replacing it. The opposite is true when we look at the working energy of each. The dishwasher has a relatively resource intensive operation using water, electricity and chemistry to clean the plates. Paper plates can be used and then composted.
There are a lot of variables that could change the net environmental impact of each one. That however is a rabbit hole for a more enthusiastic commenter.
My favorite home is net zero energy and I try to run it net negative for all inputs including food, trash, fuel, power, water, wastewater, and money. I have a dishwasher, but I've never used it and probably never will. As long as it remains unused, it registers zero for all inputs except trash for which it scores negative. Although it's in nearly new condition, it represents negative trash because it was salvaged from a high end remodel. My main reason for not using it at all is that it takes too much power, but if I did use it the power would be generated by the home and the waste water would go directly to the plants.
Likewise I never use paper plates for eating on. I do use them for compost and that helps me to maintain negative trash and food targets, but I never buy them or use them for eating. The main reason for this is that I am not able to calculate the impact. Even if I could work out the numbers it would be positive for water, energy and money. Paper takes a lot of water to produce, energy to transport, and money to purchase.
I realize that this answer doesn't fairly answer the question, but it explains the reasons for the strategy I've chosen and it certainly makes the calculations possible. If I absolutely had to choose between the two options you offered, I would run the dishwasher at a time when I had a lot of surplus power for the reasons stated above, but the equations would be much different for people who purchase dishwashers and haven't closed the loop for power, waste, money, water, and wastewater.
It might be interesting to note that I sometimes use banana leaves as a substitute for paper plates, and as compostable cookware that doesn't require washing at all, but I'm not sure if that pertains to the question well enough to be included here.