Why do we check Idle operation with carb heat on during a Cessna 172 run-up?
Because, in engaging the carburetor heat, you are decreasing the density of the air entering the engine with the same fuel flow, thereby richening the mixture.  At idle power, if the throttle idle set screw is not properly adjusted, there is a risk of an engine failure from too rich a mixture entering the engine.  The idle power carb heat check is a good way to verify the engine won’t do this in flight.