Does image texture resolution impact rendering time? I'm trying to make a high-res bump map for human skin. I've seen many examples where there was quite a lot of detail on the texture, although it's mostly in portraits. I've downloaded and open the OpenEXR file that have the texture map included in it. The image resolution was at 8192 x 8192. Would such a size have an impact on rendering time, if so, does it vary by CPU/GPU? Also, will it affect the performance of the 'Texture Paint Mode' when the map is being painted directly on it?
Render time will likely be affected, but not much so, in theory it should make a negligible difference unless you are working close to your hardware limits. 

It will however have a significant impact in memory consumption. If you have little memory resources available for rendering this may have a more noticeable negative impact, like for example while GPU rendering on a card with low VRAM amount.

Texture size will significantly affect painting performance