How does graphics card's memory affect video rendering performance?
1) Everytime the GPU has to process something, it will load the necessary data (in that case your video frames) into the VRAM. Simple as that, the GPU cannot work with your system memory.
Though the GPU gets it's data from the system memory, so your system memory will act as a buffer and you don't need to load that much data at once into the VRAM.

Though programs which offer a real-time workflow will want to load as much data into the VRAM as possible. That's the reason why you can allocate a specific amount of VRAM in Adobe After Effects (not 100% sure if you can do the same in Premiere).
So every time you apply an effect in Premiere that is GPU accelerated, the GPU will have to request data into the VRAM to process the frames.

New versions of OpenCL and CUDA are improving the way of how the GPU will get its data and reducing the overhead caused by the CPU which will reduce the amount of VRAM needed at a time but VRAM will always be a very important factor with the current PC architecture.
Current GPUs use GDDR5 which is A LOT faster than DDR3 system memory. This is needed as the memory speed is a very important performance factor in GPU computing. Overclocking your VRAM can give you quite a performance boost in memory intensive 3D rendering applications (though probably not in Premiere).
Current GDDR5 based GPUs like the GTX 780 Ti are capable of more than 300 GB/s (336 GB/s in this case). DDR3-2133 is substantially slower with a peak bandwidth of 16,66 GB/s per module. Also 2133 is rather uncommon in OEM PCs, DDR3-1600 and 1300 is more common.

2) So you will benefit from a large amount of VRAM everytime you work with very high resolution images or many videos at once aswell as with high-frame rate footage. Premieres rendering engine is utilizing the GPU for both the preview aswell as the final rendering if you chose so.

In practice I'd say it's nonsense to have more than 2GB of VRAM for Premiere unless you work with other applications at the same time that utilize the GPU. Processing bitmaps doesn't require all that much data at once unless you work with crazy resolutions like 8k or 16k, the processing capability of your GPU will be the bottleneck before your VRAM.

Thats my opinion based on what I know about the Premiere video engine and GP-GPU. Premiere might do some other stuff in the background that requires more VRAM. So if you have a very specific use case I'd rather make a benchmark or search for one before making a purchase decision. If you just want to make a general guess this information should help you quite well to make a decision.