The stickers will survive a non-aggressive washing, I have done several. The big IBM stickers, that is. The ancillary paper stickers without plastic coating are weaker, but it can still be done.
Clean the case back of all big stuff and junk like black marks (try a pencil eraser) and adhesive residue (organic solvent such as citrus oil), until you are down to "just plain old dirt" for the final wash.
Wash with warm soapy water in the kitchen sink and rinse well. You will see a discolored trace around the edge of the sticker where the paper under layer is wet below the plastic upper layer, but this will go away.
I would advise you not to scrub the sticker, and make the sticker area the last to be soaped and the first to be rinsed. You can work your way around everywhere else to keep the contact time between the sticker and wetness to a minimum.
I use dish soap for this, and laundry soap with "oxi" added for everything else, including keys.
