Do diamagnetic material exhibit induction?
Diamagnetism in ordinary materials (not supercondutors) is a relatively weak effect.  Whether a current is induced in a material by a changing magnetic field depends on the material's conductivity.  The diamagnetism will appose the external magnetic field somewhat, but since this effect is weak the magnetic field will still penetrate and therefore cause a current if the material is capable of conducting a current.  Note that copper is a diamagnetic material.

Superconductors exihibit what appears to be strong diamagnetism from the outside point of view, but they actually do this precisely because the change in the external magnetic field will cause a current.  Since the superconductor has essentially 0 resistivity, that current continues to circulate instead of die down like it would in ordinary materials.  As a result, the superconductor builds up a reverse magnetic field to the applied field, acting like a strong diamagnet.  In actuality it is acting like a perfect coil.