Horse jumping (for height) is no longer a widely practiced sport, as it is considered exploitative and dangerous to the horse, so the official world record hasn't changed since 1949.
That record was set by a 16-year-old horse named Huaso, who jumped 2.47 m (8 feet, 1 1/4 inches) at a "jump-off" in Vina Del Mar, Chile, on Feb. 5, 1949. He was ridden by Captain Alberton Larraguibel Morales, but I think it's fair to say that Huaso did the hard work.
You can see a video of the jump here.
Huaso's record was unofficially broken by a horse named King's Own, who jumped 2.53 m (8 feet, 3 1/2 inches) in front of a crowd of 25 people and some movie cameras. This record was not considered official, however, because it was not set at a public gathering.
Incidentally, the human high jump record is almost identical to the horse record. Cuba's Javier Sotomayor, the most dominant high jumper in history, cleared 2.45 m in 1993 at a meet in Spain and his record has stood ever since.
See: The Ultimate Horse Site, The Physics of Riding Horses