If I used hand dyed fabric or floss, I try not to wash the piece because I'm afraid the colors will run. For those, I just try to be clean as possible. For other pieces, I soak it warm water with woolite or just a bit of baby shampoo
Ideally you'll want to wash things before even working with them if you suspect that colors can bleed. Specially if it's hand dyed. This way you can see how much of the color is lost or if the slight humidity can make part of it bleed into something else.
Sometimes when the case is bad enough even contact with sweat can ruin something. Other times a dark hand dyed thread can bleed into a lighter background out of nowhere, or vice versa. I remember my grandma having some hand dyed kitchen towels with some embroidery and after a while you almost couldn't see the embroidery anymore due to the darker fabric bleeding into it.