I asked a physicist this question. Here's what I understand.
First, Ohm's "law" only applies to ideal resistors. A superconductor is a nearly ideal inductor. That means that its voltage and current are related (to a first approximation) by the equation v=L*di/dt, not by Ohm's law.
Faraday's law still applies to superconductors. A change in magnetic field causes a voltage to appear in the superconductor. This voltage causes current to flow. Because voltage is proportional to the derivative of current, a transient voltage (when integrated) results in an enduring current in a superconducting loop. The current will stick around as long as the magnetic field is present. As the magnetic field is being removed, a voltage transient of the opposite polarity appears, which induces a current in the opposite direction, and when the field is gone, the current in the loop is 0 again.
This voltage is in fact what causes the current to flow -- it's not possible for the current in a superconducting loop to change without some kind of voltage present. When the current is constant (dc), as in a perpetual current loop, the voltage is 0; v=L*di/dt is satisfied.