Why does using carburetor heat increase fuel consumption?
I can think of two reasons:

The efficiency of heat machines depends on the difference between lowest and highest temperature in the cycle, relative to the highest temperature, as formulated first by Sadi Carnot. Simply put, the efficiency cannot be larger than $$\eta_{max}≤\frac{T_{max}-T_{min}}{T_{max}}$$
where all temperatures are expressed relative to absolute zero. If the entry temperature $T_{min}$ is raised by carburetor heat, the value of $\eta$ will drop. For the nitpickers: Piston engines are best described by the Otto cycle which uses an isochoric addition of heat and is a bit more messy to describe than the ideal Carnot cycle. If you want to include Diesel engines, use the Trinkler or Seiliger cycle.
With the higher entry temperature, all temperatures in the cycle are higher and you need to lean less in order not to exceed the desired cylinder head and exhaust gas temperatures. More fuel is needed for cooling, so the richer mixture will drive up fuel consumption.