Not quite. Leakage current in CMOS circuits became the dominant source of power consumption around the 90 nm and 65 nm nodes, long before quantum tunneling was a major factor, and often exceeded dynamic switching power. This led to the introduction of multiple threshold-voltage devices and body-biasing techniques to dynamically adjust Vt and curb static leakage.
Well, until we scaled transistors down to the point where electrons quantum tunnel across the junction. Now they're leaky again.