It should work fine but yes, you would need a heavy-duty, over-sized suppressor, and you may have to water-cool the thing to counter heat build-up. The biggest drawback would be the noise from the mechanism. A minigun is going to produce a certain amount of racket, no matter what you do.
The gun rotating cycling shells is louder than the shot. A minigun for all practical purposes suppresses itself as the majority of noise is from the rapidly expanding gas as the bullet leaves the barrel. The minigun is typically spinning fast enough to disperse that gas so you don't get much boom. That has the same effect a suppressor does.
The end result would be about the same effectiveness as putting a second suppressor on a suppressed rifle. Pretty much nothing.
"Gun control: The theory that a woman found dead in an alley, raped and strangled with her panty hose, is somehow morally superior to a woman explaining to police how her attacker got that fatal bullet wound." — L. Neil Smith
The problem with being stupid is you cannot simply decide to stop doing dumb things...
"I crapped my pants to avoid the draft!!" -Ted Nugent
If it was desired a 16" battleship deck gun COULD be suppressed. I don't know what kind of decibel drop you'd get and the sonic boom of the projectile would still probably hurt your ears. At some point the exercise becomes useless.