There's some truth to this, but it's not really a paradox.
First of all, and most importantly, water is always evaporating. The Sun's heat will make it evaporate quicker, and a body of water with a larger surface area will evaporate at a greater rate, but liquids evaporate anyway.
In addition to this, in the real world there would also likely be further transfer of heat from the Sun through the medium of the cone, and the air above the water, heating up the now smaller quantity of water which would thus require less energy to achieve a given temperature increase and therefore a given rate of evaporation.