Summarize:

If you have dead skin or cuticles that changed colors because of the dye, use a cuticle clipper to carefully remove the skin. This will save you from using harsh nail polish remover on your skin. If you've tried everything and your nails are still discolored, your best bet is to cover them with a fun nail polish. Be fashionable and disguise the stain at the same time!
Trim your cuticles if they are dyed. Paint your nails if you just can't remove the dye.