Mak,
The fuel flow into the carb is dependant on the volume of air passing through it. The volume of air, for any given throttle setting and engine speed, will not change if you apply carb heat. Therefore, the fuel flow will be identical.*
What does change, though, is the density of the air. Warm air is less dense, therefore there is less mass of air going into the engine (or, to put it another way, fewer molecules). If there are fewer molecules of air, then you can't produce as much power, since you need the air for the fuel to burn.
As a side effect, because there are fewer molecules of air but the fuel flow is unchanged, the mixture will be richer. This is why it is possible that you will get a rich cut if you add full power suddenly with the carb heat on (e.g. in a go-around).
Hope that answers the question,
FFF
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* Note the slight simplification - fuel flow will actually only be identical if the RPM is unchanged. Because the RPM drops slightly, you will use very slightly less fuel, but this is pretty insignificant