How much can the CPU affect the Performance of a Graphics Card

Solution
Well, if you pair a very low end CPU (lets say Core 2 Duo from 8 years ago) with a high-end GPU (like a GTX 980) the processor will limit the performance of the GPU.

So, this all depends on the specific parts. But to answer the question, the CPU can cripple the performance of the GPU if they're not spec'd correctly.
Well, if you pair a very low end CPU (lets say Core 2 Duo from 8 years ago) with a high-end GPU (like a GTX 980) the processor will limit the performance of the GPU.

So, this all depends on the specific parts. But to answer the question, the CPU can cripple the performance of the GPU if they're not spec'd correctly.
 
Solution
("Spec'd correctly?)
Technically, one or the other is usually the bottlle-neck.
Generally folk agree that GPU is usually the weaker link.
In my experience, upgrading CPU by a good step made a nice difference. (Not recently. From AMD Athlon 630 quad to Phenom 965.) But the game I play most (ArmA) is known to be CPU intensive.