Do hurricanes cool the atmosphere (by taking heat out of the ocean/air and using that energy to essentially power the winds)?
We believe hurricanes form due to thermal inhomogeneities in the atmosphere, similarly to whirls in water being heated on a cooker. The lower parts of the atmosphere are warmer due to heat received by conduction and evaporation, both from the ground and from the ocean. Some energy initially absorbed by the ocean (water absorbs visible light) thus later indeed goes into the atmosphere and heats its lower parts. Since the air expands upon heating, it gets lighter and may move upwards, which creates macroscopic flows of air. These flows of air then lead to storms, and with help of the rotation of the Earth, on larger scale (>1000km) may attain strong vortex character and lead to hurricanes.
So the creation of the hurricane and air flow in general can be said to partially extract energy from the ground and thus has some kind of cooling effect on it. But when the flow is already present, it also gives energy back. When the atmospheric air moves violently above the ocean, it moves the ocean water. Due to internal friction in the latter, the atmosphere does some work on the water. As a result, atmosphere loses some energy (and would lower its temperature if this was the sole mechanism occuring) and the ocean receives it.
Also during the storm the rain water may be falling into the ocean which means the atmosphere gives energy to the ocean. On the other hand the ocean evaporates into the atmosphere at the same time, which brings energy to the atmosphere. There are many such processes with all two effects (refrigerating and heating) possible occurring at the same time, and it is hard to say what will be the net result. It is perhaps fair to say, that on average after long time (1 year?) the atmosphere and the ocean are in dynamic equilibrium that may be shifting slightly.