Yes. Horses absolutely can get sunburn. They generally get it on their nose, and sometimes the lower legs, but only where they have white markings. The pink skin lacks melanin and thus is vulnerable in the same way redheads are vulnerable. (Grey horses have dark skin and are not particularly vulnerable.
And, just like in humans, repeated sunburn can increase the risk of skin cancer.
Sunburn can be prevented with sunblock - there are special horse sunblocks available but a lot of people just use pediatric sunblock, which is cheaper. Sunburn can also be treated with zinc and castor oil cream, and in severe cases with antibiotics to prevent infection.
Sunburn can be a sign of photosensitization, which is usually caused by eating too many buttercups or clovers.