Hair dye can stains are some of the most difficult stains to remove depending on the surface. Hair dye on carpet will be there forever, but on linoleum it's a simple fix with the cleaners and some elbow grease. It's much easier if the stain is fresh, but even if it isn't it should still come off with a few household cleaners and determination.
Apply chlorine bleach to the stain pouring just enough to cover it. Allow this to sit from 30-60 minutes and wipe away with a cloth or paper towel. This is not a good idea on coloured linoleum as it will bleach out the colour of the tile as well.
Apply nail polish remover or acetone to cotton balls and scrub the stain out.
Dampen a Mr. Clean Magic Eraser and scrub the floor until the stain is gone.
Pour white vinegar onto the stain, being sure to cover the stain completely. Wipe up the vinegar in about 30 minute with a paper towel or something disposable.
Use extreme caution in handling hair dye which has a tendency to stain everything it touches. Lay down newspapers or plastic over the floor in the future to prevent spills and stains.
Avoid the use of harsh chemicals as they can destroy linoleum.