Some wires can be removed by bringing in pixels from either side of the wire, ala healing brush or whatever they want to call it. This works on mainly very thin, true sticks and wires.
In general, you're going to be erasing your rig from the top layer, revealing a clean area in a layer below. How you get that clean area differs. Clean plate is ideal, an offset from another frame elsewhere in the sequence and tracked back into place is second... A 'paintfixed' hero frame is next on the list. Then you're getting into
combos all the way down to a frame by frame, manual reconstruction.
It works best with locked down shots from a tripod or stationary camera person. It doens't need to be still but it wont work from a moving dolly shot or camera crane...
Basically you take a still frame of the background where the wire isn't in view or even if you paint it out in one frame and use the still frame to cover the later shots.
The problem with the paint tools in AE is that they are difficult to use and ofter times a tracked frozen frame works best to hide wires and stuff. you can always use the match grain to add back some noise.