Time and time again I've seen people post that the dye from their threads has run, ruining a project. I've written this post in the comment section of numerous blogs. It occured to me that I should just write the post once and refer people to it! (Smart girl.)
I developed this specialized knowledge when, after stitching Heart in Hand Wee Baby, I washed it. I had forgotten I had stitched with overdyeds (claret!). D'oh!
I was told that really cold water would (eventually) remove the dye from the places it shouldn't be. Unfortunately, it was summer in Los Angeles and our apartment was not air conditioned. Sticking a piece in cold water was going to be difficult. My first thought was to add ice to the water. So I put the stitchery into a brand-new dishpan with iced water. Left out on the counter, the water was warm in no time. So I put it in the refrigerator.
I changed the water twice a day. And after about a week, the dye was gone! It was a miracle. Actually it was science. I can't explain the science to you, but that doesn't mean it's not science.
Use this knowledge as appropriate.