Horses can indeed get sunburn. It's more likely on areas that have thinner fur, lighter skin and both - the areas most at risk are the nose between the nostrils on horses with white faces and the area around the eyes.
It shows up the same as sunburn in humans - a reddening and flaking of the skin. And its prevented the same way.
Equine sunblock is available - but human sunblock is cheaper and just as effective. A fly mask or fly veil can help keep the sun away from the horse's eyes (and also protect them from insect irritation).
Treating sunburn is also easy. Aloe sunburn treatments are effective. A traditional remedy is zinc and castor oil cream, which can usually be found in the pediatric aisle of the drug store. Zinc and castor oil cream both treats existing burns and helps protect the skin from further damage.
This mare's white face puts her at particular risk for sunburn.