I don't think it's a bad idea. I don't think everyone would do it but if you say something like "In an effort to be more Green please consider bringing re-usable table service from home." I think having small "washing" stations set up for people to use would be a nice idea that way you don't have to carry dirty plates back home with you.
I love the idea of trying to encourage families to be more environmentally conscious. It certainly can't hurt to suggest families to bring their own from home. Not everyone will, but cutting back on the trash you are creating sounds like a positive to me. Hopefully the idea would catch on and spread.
Just my $.02.
I'm with Rockne. Although as much as you want to be green sometimes you have to pick your battles.
If there was a way to wash them onsite and either have them take them back home or donate them to the Salvation Army or Goodwill.
I would probably do the same thing (bring paper and plastic), but maybe if you stressed the "green" part of it and specifically mention bringing real plates and silverware, people may actually do it.