If I change the phase of a current running through a loop of wire, does the magnetic field produced by the loop change its phase by the same amount?
In general, yes.  The magnetic field is proportional to the instantaneous current, so if you change the current waveform then the magnetic field will change in the same way.  Note that if there is a magnetic core inside the loop, then it is possible for the core to saturate, which will change things.