I have worked with smaller sized TFTs and may not be able to best answer your question.
Yes, the fast refresh is a function of the processor you attach to the TFT. In my experience, I have had reasonably fast refreshes, but, nothing approaching video frame rates yet. There are plenty of projects on the www that can do those frame rates, but, you need to have a lot of patience and tweak the code in assembly.
If you intend doing it with a processor, consider using one with a parallel 16 bit external memory interface. That can help quite a bit to speed things up.
As for the cell phones and their video capabilities, these days phones use dedicated chipsets which are able to refresh the screens at quite a fast frame rate. Another possibility is to use a FPGA to update the TFT. Doing it all in code run on a small processor can be quite exciting and challenging.
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