Passover brings an interesting quandry this year- we go through a great many Kosher for Passover dishes at the Lail Pesach seder on Friday night, and then have a couple of meals on Saturday. Given that A) it is forbidden to prepare for a weekday on Shabbat, B) it is forbidden to prepare for the second day of Yom Tov on the first day of Yom Tov, C) handwashing dishes is pretty universally prohibited on Shabbat and Yom Tov except for immediate use and D)the second seder should begin at Sundown on Saturday night and requires a clean, set table- assuming one doesn't have unlimited quantities of Kosher for Passover dishes- when would one be halachically permitted to was the dishes so as to prepare for the second seder?
You're definitely not the only one in this "quandary". First of all, I believe that there is some leniency that allows washing dirty dishes if the dirt is either "offensive" or one is troubled by seeing the dirty dishes lying around. I have to research this, but, from what I recall the decisive factors are not as simple as you would think, since, there is always that element of hachanah (preparation) for the next day.
So, I'll address the more "practical":
Most people that I know that use "real" plates deal with this. Preparations begin only once the 2nd day of Yom Tov begins. This is not as terrible as it seems, actually. In most cases, the men are still in shul, because many shuls aren't starting Ma'ariv until exactly when Yom Tov II starts. Even the places that do start a bit early - by the time you say Halle and get sefirah in at the right time and walk home, it's about 30 minutes after the 2nd day has begun. That should allow plenty of time to rinse dishes and set the table. If you have a "large staff" of helpers at home (large family plus some guests), it can be done fairly easily.
If you're really concerned, though - Either have two sets of plates around or better, as I have done numerous years, paper and plastic really help out for this situation and there's nothing to wash afterwards. Trust me from personal experience. I spent numerous Pesach Seders staying up until 2 - 3 AM washing dishes and putting away food. (Incidentally, at night, it's not a problem to wash the dishes that I will be using for lunch the next day.)
Mo'adim Lesimcha to all...
I discovered this article which mentions a leniency. So, while the seder(im) are over, these rules would apply to any Shabbat or Yom Tov.
If the dishes are required for use on Shabbos, it is permitted to wash them.
According to some authorities it is permitted to wash dishes even for the purpose of having the house and kitchen clean and tidy.
Dishes must not be washed with sponges that involve a concern of squeezing out water. Synthetic (non-absorbent) material can be used, or just use your hands. It is permitted to use detergent.
The Shulchan Aruch (OC 323:6) and the Mishnah Berurah (323:26) write that it is prohibited to wash dishes if they are not required for used on Shabbos. However, if the household norm is for the sink to always be clear of dishes, and the sink being full of dishes is seen as clashing with the honor of Shabbos (or Yom Tov), then it is permitted to wash the dishes.
The source for this leniency is Rashi (Shabbos117b), who writes that the prohibition of washing dishes is on account of preparation for weekdays. This implies that if the washing is not for preparation for weekdays, the act would be permitted. [There is certain debate about the position of Rambam in this matter.] The Maharshag, Orach Chaim, Vol. I, no. 61, writes explicitly that this is the case, and permits washing dishes for the sake of hygiene and cleanliness; Tzitz Eliezer, Vol. 14, no. 37 concurs with the ruling.