Clocking and changing register states requires charging and discharging the gate capacitance of a bunch of MOSFET transistors. The current that results from moving all that charge around encounters resistance, which converts it to heat. Silicon is only a "semi" conductor after all.

You are correct that there is energy bound in the information stored in the chip. But last I checked, our most efficient chips (e.g., using reversible computing to avoid wasting that energy) are still orders of magnitude less efficient than those theoretical limits.