If we turned the Earth into a black body and used all Solar radiation to run computers or move stuff around at the theoretical limit, we'd still need the surrounding space to radiate off the heat as our cold well. So the temperature would be at whatever that equilibrium condition is, but wouldn't steadily increase. The equilibrium condition could be all over the place and would be largely determined by the composition of the atmosphere.

For the record I'm not in favor of paperclipping the planet like this. If anyone was wondering.