Actually, neither fat nor muscle tans, only your skin can tan; your skin will tan the same whether it is over muscle or fat.
When the Ultra Violet rays from the sun (or specially designed lamps) reach your skin, the melanocyte cells in the skin produce melanin. The melanin is what causes the pigment, or darkening of your skin.
Melanin is your body's natural defense against burning, the pigment absorbs the UV radiation, protecting the other cells in your skin.
Care should always be taken when exposing your skin to the sun; a new study by the GW School of Medicine and Health Sciences finds that UV-induced melanin may be directly involved in the formation of melanoma, the most lethal type of skin cancer.