We all know current will not flow until unless the path doesn't get complete. In your case when you pass the current in the zero resistance wire. The data which we will be reading at the other end of the resistor will be voltage. So this scenario only works for finding the unknown resistance value. If you are not reading or connecting the zero resistance wire. Then nothing flows through that. Because the circuit doesn't gets complete.
In your sceond case, It is almost something like you are shorting the two terminals of the voltage source. You will knowing the scenarion what happens if we short two terminals of voltage source.....