My girlfriend was dying her hair (the permanent kind) and ended up getting some on the wall. However, I didn't find out until it had completely dried and now soap and water is not enough to remove it.
Obviously I don't want to use anything that will damage the wall or paint, and It's not worth buying paint for the wall.
What is my best option for removing hair dye from my wall?
The only option I can see short of high-filler sealing primer and then paint might be to bleach the dye using peroxide bleach similar to what your girlfriend most likely used on her hair before the dye step. Peroxide is a very powerful oxidizer, and will denature most dyes. It's likely to damage the paint, however, so if you rent, you may as well just leave it as is -- it'll cost you the same from your damage deposit when you move out either way.
Press on the tape so that it firmly sticks on to dye part.
Leave it for 3-5 seconds
Then try pulling the tape back suddenly
For sure, the tape will remove the die part along with very light amount of paint.
You can repeat this until you are satisfied with the wall
Anyway, if you are not satisfied with cleaning the wall using powerful chemicals that do not lose paint, you will have to suffer some paint loss. Moreover, paint samples are available in today's market that costs only very less price, you can purchase that and paint the lost portions.