How much hay do heavy horses eat daily?

As with any horse, the daily intake is based on the horse's weight, plus how the horse is being used. On the average, horses need 1% of their body weight in forage to maintain a healthy body mass. As an example, my Arab, who weights 900 pounds dripping wet, eats less than ten pounds a day in grass hay. As we start putting on miles, or the weather gets cold, her food needs go up from there.
As the draft (heavy) horses go, the rule of thumb still applies. Keep in mind that this is merely a starting point, and needs to be tailored to the specific horse and its environment.
One thing you should get is a "tape measure" for monitoring your horses weight. They're easy to use; merely wrap it around the horse's girth where the heart girth would sit, then read the weight on the tape where the end lays against the tape. It's not a perfect weight, but it's great for keeping track of weight fluctuations.