How do sound-dampening drywall clips affect the ability to hang objects on the wall?
The sound dampening system you describe builds a metal frame rigidly attached to clips that have rubber isolation bushings. There are two possibilities in attaching items of significant weights.

Directly to the metal framing

This would spread the load over the entire metal beam, much like attaching it to a steel stud. It would not be quite as supportive since the beams are not directly attached to each other or to a sole and top plate. 

This approach would probably be sufficient for almost any load a conventional stud mounted approach might support short of things that involve dynamic pressure, such as a punching bag. As usual, mor attachment points spread out the load and reduce chance of failure. I would at least double the number of mounting screws/bolts. 

However, such a  mount might slightly compromise the acoustic benefits since the object would be rigidly attached to a steel member which could transmit some vibrations. Less of an issue with object that do not vibrate or emit sounds. You also need to ensure that you hit the steel beams but miss the studs and clips. If you hit the stud with a mounting screw, you defeat the clip's isolation function.

Using heavy duty drywall anchors

There are several possibilities discussed in this answer. I would tend toward the newer toggle type anchors that use a 3 inch metal bar to spread the load behind the drywall.  These are probably the strongest anchors after direct stud mount. Using a toggle also isolates the load from the steel frame. Drywall is less resonant than steel, so there is less chance of compromising the acoustics, but the panel may act as a bit of a drumhead.

You might want to think of ways to free float heavy objects in front of the wall, on an easel or something similar.