Which breed of horse is black but turns white when it gets older?

No breed of horse is exclusively black, there is one breed, the Friesian that comes in mainly black, but they also carry the gene to express red (chestnut.)

The Extension gene is responsible for black and red pigment. If a horse is Homozygous (dominate) for black then it will have EE, if the horse is Heterozygous (recessive) for black it will be ee and will be chestnut in color. It is also possible for a horse to be Ee, that means it carries one copy each of black and red and can pass either one of those along resulting in either black or chestnut foals if paired with another horse that is Ee or ee.