When you're setting up your shots, pay careful attention to where the shadows are and think about how you're going to erase them. What you want to do is adjust the rig and the lights until the shadow moves to someplace where it's not going to be a problem.
Yes, that's a common problem. When you erase the rig to see through to the clean background, something is different. Maybe the puppet cast a shadow behind the rig, but in the clean background shot you took without the puppet, there was no shadow. Or maybe the camera or set moved, so things don't line up perfectly. Back when I shot on 16mm film, you could always see where the rig had been removed when you played it back, because of a slight movement in each frame, and because the grain was different for every frame. With a clean, noise-free digital image, it should be possible to erase wires without it showing, but you have to watch out for any changes in the background behind the rig.
I take a clean background shot at the beginning, and another one at the end, in case anything has moved or warped over the 3 to 6 hours it took to do the shot. And sometimes I use another frame of the animation, maybe one or two frames earlier or later, as the background instead. One way is to just have a second layer with the whole shot in it, and try sliding it forward or backwards a couple of frames. Often that's enough to put the wire in a different place.
As Strider says, you have to look out for what will be behind the wire when you are setting up the shot, to see if you will have any problems.
I have a creature landing on the ground with a wire supporting it to animate today. I will have to erase the wire, and the shadow of the wire as well. But I need the shadow of the puppet to be there, that's why I am shooting it on a set and not just greenscreening it. I'm still setting up and checking where the shadows will be.