Does the video card influence the speed at which a video is processed and exported from Adobe Premier Elements?

This depends on which version you're using. I know GPU acceleration is supported in the newer CS4 products.

From the above link:

Note: For details regarding GPU support of individual Adobe products, please visit that product's system requirements page on the Adobe website.

They also show a list of compatible video cards. By entry level, it depends on if you're referring to desktop GPUs or workstation GPUs. Typically workstation GPUs are much better at this sort of thing. An entry level workstation GPU on their list would be the Quadro FX 1700 or the FireGL V-Avivo 5600.