It's most apparent when the four main characters are entering the tunnel to visit the wizard. They all have their arms linked--but the scarecrow and the lion linked arms in such a way as to catch the wire. Watch as they subtlely unlink arms again so as to free the wire.
The lion's tail didn't have a string attached to it (possibly in a scene or two but not the majority of the time). There was a special done on it and it was said that the lion's tale was mechanical (computerized) and so he had a remote control for it. So whenever you see it move, it's because the lion moved the remote.