Remove as much of the dye as you can from the wall immediately. Use an old towel to blot the hair dye, working from the outside of the stain into the center. Do not wipe the dye as this could cause the stain to spread.
Combine about a tablespoon of dish detergent and warm water in a small bowl. Agitate the water to create suds. Apply the suds to the wall using a sponge and gently scrub the area. Work from the outside of the stain in to avoid smearing the dye.
Apply a few drops of hydrogen peroxide to a cotton ball. Rub the the hydrogen peroxide on the spot and allow to sit overnight. Repeat if necessary.