If you dumped waste heat directionally it would act as a photon rocket.
If you somehow reflected all the light of a star in one direction, the momentum change in the light would produce thrust on the reflector (much less than is needed to move a star), and since no substance is 100% reflective the huge amount of surface area would nonetheless heat up and be visible in the infrared due to its vast surface area. The same is true of any substance directing waste thermal radiation - you can shunt most of it in some direction the material will definitely reach an equilibrium temperature warmer than the cosmic microwave background anyway.
Furthermore, all the energy would eventually get absorbed/used SOMEWHERE and become heat. Local conversion of energy to other forms would also always involve waste heat production.
As far as I am aware, black hole surface gravity is analogous to temperature and surface area is analogous to entropy when you start digging into their thermodynamics. Dumping waste thermal radiation in will make them bigger and they will eventually re-radiate thermal radiation via the Hawking mechanism in the far future.
the huge amount of surface area would nonetheless heat up and be visible in the infrared
Unless you're spending some of the energy the Dyson sphere is collecting to actively cool that surface area.
all the energy would eventually get absorbed/used SOMEWHERE
Sure, but if you point it at another galaxy, it will take a very very long time for something to happen (and in a galaxy far, far away, too).
This thread is for asking any questions that might seem obvious, tangential, silly or what-have-you. Don't be shy, everyone has holes in their knowledge, though the fewer and the smaller we can make them, the better.
Please be respectful of other people's admitting ignorance and don't mock them for it, as they're doing a noble thing.
To any future monthly posters of SQ threads, please remember to add the "stupid_questions" tag.