I saw an ad for Gumout Complete Fuel System Cleaner.
Has anyone tried this?
Does it do any good?

Gumout is not a bad cleaner.  But there are better ones.  Several years ago most of the retail mfgs of chemicals toned down their products due to liability and EPA requirements.

If you go with a professional product, you will get a much more effective product.

On the retail level, Techtron is very good.  Berrymans is also decent.

Justice Brother's is excellent.  Second to none.  But it's hard to find.  Summit Racing does have it.  Another truly great product is GM Top Engine Cleaner.  You can get at most GM parts departments, and yes you can mix it with the gas.  ZEP is also excellent, but strictly sold to professionals.

The GM Top Engine cleaner is excellent. Haven't used the Justice Bros. products, but have heard good things from others. Synergyn fuel injector cleaner is really good, also. I use this quite a bit, and have had awsome results. Whatever you use, always check/change your fuel filter(s) afterwards, as any crap the gets dissolved in the tank/lines will wind up in the filter. Gumout is OK, but, as Ken says, there are better products out there.

Thanks guys.
Any special precautions for an 82 with the Cross Fire Injection system?

Nothing special.  Just make sure you clean under both throttle plates, and all the lines between the two units are clear.

My fuel filter is in the line down below the engine, by the frame. Do I have to empty the tank when I change the filter?

No, not at all.  Just keep in mind you may have up to 15 psi remaining in the line when you take the filter loose.  Go go slowly and carefully.  Be prepared to catch spilled fuel.  Loosen the fuel filler cap before you start to relieve any minor pressure in the tank, or it could continue to force fuel out of the line when the filter is off.

Not really very difficult.