What effect does sun have on hair dye?

The main reason is that skin is alive and hair is dead. Melanin is the dark brown pigment produced by your skin to absorb some of the UV radiation from the sun.

The melanin in your hair is destroyed by sun and your hair will not produce more melanin to respond to that since it is dead. Your hair will stay lighter until new hair grows in.

Skin on the other hand is alive and when the melanin is destroyed by the sun, the skin responds by producing more melanin and thus your Skin get darker.