I know what the ATF fee to short barrel a rifle is... Is it the same to get permission to have a suppressor?
It's actually a tax stamp, looks a lot like a postage stamp, except it says $200.

They just stick it on the approved transfer form like a regular stamp.
It's $200 for an MG, suppressor, short rifle, short shotgun, or Destructive Device. It's $5 for an AOW (any other weapon) which is things like pen guns, cane guns, short shotguns whose receivers have never had a stock installed, pistols with a forward pistol grip but no stock (like an SP-89 with an MP5K forward grip) etc etc etc ad infinitum.
Identical process for all of the above. Here in Arizona, it goes more or less like this:
1. Get the three fingerprint cards and the 2 copies of the form (usually an AFT Form 4) from the dealer. Whether you pay for the item now or later is up to you and him. Most cases, you pay up front.
2. Make an appointment to get interviewed by the county sheriff or similar high-level local LE type. Sometimes your state police have an office that handles these things. First time around, you'll make lots of phone calls explaining things to various people until you get an appointment with the right guy. Your dealer might have some great advice on this front. Mine wasn't helpful. If you have a pal who has done transfers recently, he might be able to save you a lot of hassles by directing you to the right guy.
3. Take the forms and the print cards to your meeting. The forms will need a passport photo stuck onto each of them. Wear business clothes but don't go like you're dressed for a job interview. Be relaxed, and don't go armed in any fashion (no knife even) into the police facility.
4. Talk with the LE official in his office. Be yourself. You'll find that they're usually totally cool with the idea of you getting an MG or suppressor or whatever after you talk with them for a few minutes. Sometimes they might even joke around with you. Most cops are gun guys, but they're wary of everyone at first. Once they understand that you're just jumping through ATF hoops and that you're going to get the going-over by the FBI before you take possession of the thing, they relax a whole lot.
5. Get fingerprinted three times over. Make sure the prints are really nice and clear. Let your new LE friend take you to the front of the line, and impress upon the old guy doing the prints that he needs three nice sets for Mr. Gun Owner here.
6. Show the LE official your CCW. (Don't even bother with the whole Class 3 process unless you've already got a concealed carry permit from your home state: it's your bonafide saying that you're at least not an axe murderer or a complete bonehead around firearms). The LE official then runs a background check on you. He comes back and, if you've played things right, he teases you about those speeding tickets you get three years ago, asks if you work for Osama Bin Laden, laughs again when you say something clever, and then he signs your forms. Be nice, don't just run away. Chat and hang out. If the situation allows, offer to buy him a quick lunch or grab a coffee: now that you have him as a friend, you can get future Form 4s signed much more easily.
7. Take the forms and the print cards and photo copy everything. Then FedEx (not mail), FedEx them to ATF, along with your $200 check for the transfer. Include a nice clear letter explaining everything, confirming the details of the weapon being transferred, and confirming your mailing address. This is crucial, as they will never call, fax or email you with any question - if the slightest thing is wrong with the form or any of the cards, they will just mail it all back to you disapproved, after it sits on their desks for a couple of months.
8. You will wait anywhere from 6 weeks to 9 months. Most people wait about two or three months. Your paperwork will show up one fine day when you've almost forgotten about it.
9. Make 27 bazillion copies of the approved Form 4 (or Form Whatever), put the original in a safe, and have a copy of the Form with the gun at all times, like in your range bag, etc.
10. Go and pick up your nifty new Class 3 item.
Important point: you have just obtained a tax stamp. It is not a license. It is just proof that you paid tax on the gun. When you want to get another Class 3 item in the future (and believe me, you will want to), you have to begin the entire process all over again, right from step 1.