Hm, interesting. I was initially unconvinced that this could be a problem, but some back-of-the-envelope math says it's at least conceivable:
The sun deposits enormous amounts of energy onto earth every single day: Around 340 W/m² (averaged over the whole earth), or a total of 43 x 10^15 Watts. Essentially all of it is radiated back into space (mostly as infrared). We have a temperature equilibrium because energy intake is largely constant (surface/cloud albedo notwithstanding) while radiation back into space grows with fourth power of (surface/atmospheric) temperature.
Current global energy consumption is on the order of 2 x 10^12 Watts, over four orders of magnitude lower. If we somehow increase energy production by ~two orders of magnitude, to the point of ourselves emitting 1% of the solar energy intake on top, the surface temperature would need to rise by about 0.75 °C to maintain equilibrium. An order of magnitude more (i.e. three orders of magnitude above current consumption, roughly 10% of solar intake) would correspond to a 7.2 °C rise.
(Point of reference: Global power consumption has barely doubled in the past 40 years. No telling what "free" energy would cause though.)
Presumably we'd have geo-engineered a solution by that point, but it's surprisingly not too early to start thinking about the problem!
That's assuming the anthropomorphic heat is spread evenly over the earth, rather than concentrated and creating a heat island effect.
You probably can drop an order and a half of magnitude off of that number just based on concentration. And if you don't think 'free' fusion will cause us to use several times more power than we currently use, then I don't know what to tell you.
It'd be interesting to ponder whether such a "heat ray" would work, in terms of thermodynamics. Some kind of heat pump, the hot side of which is hot enough to radiate into space? I can't imagine that having a net cooling effect when considering the Carnot efficiency of a refrigeration cycle. Maybe a giant ice machine in space? (Then again, any ice would probably create more heating than cooling as it enters our gravity well or deorbits). Anyone have any ideas?
