What can be the consequences of a smaller airgap in a induction motor for the efficiency and rotor current?
If you have smaller air gap, it means that more magnetic lines from the stator are reaching the rotor. That should give higher induced voltage in rotor due to higher flux. However, it should not be much greater than with normal air gap. To current on rotor it would have really tiny impact.
Also, rotor current is a function of load. It may also be that you overloaded the motor, so it gets overheated. But the most probable cause is that heat from rotor has nowhere to go, and it is trapped between rotor and stator because of small gap. Rotor is not cooled fast enough.