Open all of the room's windows before you begin, to ensure proper ventilation. Slip on a pair of chemical gloves to protect your hands and goggles to protect your eyes from chemical splatter.
Dampen a rag with turpentine. Rub the rag over the dye-stained area to remove any residues or waxes. Allow the wood to air-dry.
Brush lacquer thinner over the affected area to remove the finish. Work the stripper into the wood using a 000-grade steel wool. Wipe away the lacquer thinner when the steel wool reaches the bare wood.
Bleach the hair dye stain using oxalic acid. Mix 5 ounces of oxalic acid crystals with 1 gallon of water in a bucket. Brush the bleach mixture over the stain.
Leave the bleach on the wood for approximately two minutes before removing it with a water rinse. Allow the wood cabinet to air-dry for 24 hours.
Dip a clean rag into a wood stain that matches the original color of the cabinet wood and rub it over the bleached area. Allow the stain to dry for 24 hours. Brush on clear polyurethane that has either a mat or gloss finish that matches that of the wood on your cabinets. Leave the cabinet to dry for an additional 24 hours.