If Stark moved to Gotham, he'd spend half his time going after global threats like Vandal Savage and the other half cloistered in his laboratory inventing things. If he has a little spare time, he'll go on a few dates or stop a random bank-robbery. He's not going to even try to clean up Gotham, because that's just not his wheelhouse. Even if his house is in Gotham, he likely wouldn't spend a whole lot of time actually in Gotham.