Watching a recent episode of Avengers Assemble it features the Destroyer in the hands of Doom, and of course the Hulk goes up against him--and gets his tuchus kicked.
However, in the comics, I think this would be a much more fair fight. The Destroyer is essentially mindless, and the Hulk does get stronger the angrier he gets.
So, who wins this close one? The Destroyer is truly tough, but I'm thinking the Hulk outsmarts him at some point.
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In a comic, the Hulk would win, because the writer would make it happen. Or he'd lose the initial fight and then win the rematch.
The Hulk is probably faster, and that would be an advantage to him.
He would likely try all kinds of different ways of smashing the thing. If e.g. they were fighting in an underground vault in which it had been stored he'd first hit it himself, then hit it with whatever was handy, then bring the entire cavern down on it. If the body of the spirit animating the Destroyer were on-site there would be a danger of its being destroyed in the mayhem. Would that kill the animator's spirit? Even if not, if the animator were an Asgardian, it might place the spirit in the power of Hela.
The Hulk could probably get the Destroyer off the ground and throw it a long way. There might be something handy he could throw it at that would stop it. Say, they might be fighting on Skipbladnir as it travels through space, and he might throw it towards a black hole.
There's significant danger for the Hulk. He'd probably fail to realise that the Destroyer's disintergrator beam could hurt him before he'd been hit by it. ("Stupid glow can't hurt Hu-AARRRGG!!") But if he were lucky he might get away with e.g. only losing an arm.
There's probably stuff on Asgard that could suck the animating spirit out of the Destroyer. Or Kamo Tharnn might owe the Hulk a favour, and be able to do that with his Runestaff.