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
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.