Frontal inversions occur where a mass of cold air moves into a region that was previously occupied by a warm air mass. The cold air, being more dense slides in underneath the warmer air lifting the warm air up. This results in the warm air mass overlaying the cold air, thus an inversion.
Any local heating of the cold air will cuase it to rise but the lifting will not continue above the frontal inversion unless the rising air mass is warmer than the overlaying warm layer.