A bit of context: I've got two graphics cards (not in the same build, my motherboard cannot support that), a GTX 1060 and a GTX 960. I use the 1060 for most rendering and gaming and such, but the 960 is just collecting dust. I'm gonna get a PC from my school that I want to install the 960 into and use it as a rendering machine, but of course since it's a school PC the processor will be pretty bad. Will this low-end processor bottleneck my 960's rendering significantly? Or will the difference be minimal?
No. GPU rendering is just that. CPU will have little to no effect - pretty sure. I am a little hesitant if system memory plays a role if you run out of graphics card memory, that could be an issue, but I'm not sure about that.