When carb heat is applied, the mixture enriches due to hotter and less-dense air mixing with the fuel. But a richer mixture has the side effect of cooling the engine down. So why does carb heat cause the engine temp to rise?
Switching Carb Heat to ON sends air to the carburetor that was heated by being near the exhaust pipe. While it is correct the fuel/air mixture will be richer, the amount of fuel entering the engine does not increase. A slight drop in RPMs may have a some cooling tendency but the heating of the incoming air off-sets this. Gasoline absorbs much less energy during evaporation than water$^1$.
Richer mixtures do burn with a lower temperature, but again the heating of incoming air results in a net gain of engine temperature at the cylinder head.
$^1$ water/methanol injection can be used help control engine temperature, but may be corrosive to engine parts