It's true that darker color skin is more resistant to burning, but as Open Mind said, they can still get sunburned. But it doesn't have to be a burn for it to be damaging. A tan is your skin's way of trying to shield itself from harmful UV rays. Skin cancer in dark skin is less common; however, they can still get melanoma, especially in the nail beds and on the soles of hands and feet where the skin is often lighter colored.
Absolutely-everyone of any skin color should be applying sunscreen daily and avoiding excessive exposure. Your skin will thank you as you get older by not having as many precancers and cancers; you'll also look so much younger than your sun-worshipping peers. Tanning is the quickest way to look old and weathered.