Not in the same way we do. High cholesterol in horses is not a condition of itself but a symptom, usually of something serious. (Foals under three months naturally have elevated cholesterol, probably due to rapid growth and chemicals in their mother's milk).
High cholesterol in ponies, miniatures and donkeys is a symptom of hyperlipemia syndrome, which is associated with obesity combined with stress) - but it's often not a noticed symptom as the condition is also associated with more obvious symptoms such as depression, fetid breath, diarrhea and colic.
Bile duct obstruction and liver disease can also cause high cholesterol in horses, as can sepsis. But the kind of chronic high cholesterol we see in humans doesn't exist in horses because of their diet.