Just to clarify, the reduction occurs because the molecules/atoms of a dielectric will polarize within the material. This polarization will create local electric fields that will counteract the applied field. Over the bulk of the material, this results in a reduction of the overall applied field. However, it does not result in a reduction in the stored energy because the energy lost in the applied field is now stored in the polarizations in the dielectric.
The production of heat would require continual agitation of the molecules. A capacitor, under DC currents, would orient and polarize the dielectric's molecules once (and in an ideal capacitor we do not consider it consuming power under AC or DC currents either) and thus not allow for a continual production of heat. However, there is the chance of dielectric breakdown if you have a large enough applied field. A dielectric experiences polarization of its molecules and under a large enough electric field this will result in the electrons being stripped from the molecule/atom, making the material locally conductive. Once a path of conductive material is created in the dielectric, a current can arise which will discharge the plates. Once the plates are discharged, there is no longer an applied electric field (or a large one) and the dielectric can sometimes resume it's normal properties.
A classic example is lightning. The electric field between a cloud and the ground becomes large enough to make the air conductive, turning it into a plasma, along the path of the discharge. This can often permanently damage the dielectric in a capacitor (even in air the oxygen often turns into ozone, but ozone is fairly unstable though).