That’s not how it works. With conservation of energy, all the energy coming in to power the computers has to be emitted somehow. Powering computers doesn’t get rid of the energy, it just makes it unusable and converts it into heat.

Right and that's why the heat is the problem, in space.

But if the collected heat comes from a large area of solar-cells, and is then focused on the small area of a computer or graphics-card, that computer might melt.

This is probably a stupid question but is it possible to recycle the heat back into electricity and use it again?

In principle heat can be converted to work if there are two bodies at different temperature. In space heat can only be irradiated away since there’s no matter available for conduction, so I would think that eventually all of your machinery would thermalize with the environment and the boundary conditions. While there should be a temperature gradient between the side exposed to the sun and the one opposite to it, I’m not sure how much of this is actually harvestable.

Yeah, that's my point.