You should look at Mike's (mike's electric stuff) videos on the iPod Nano displays. He's essentially streaming data from a SPI flash chip to a 240x240 display using a small FPGA and a PIC32.
But let's go with a micro. We'll assume you run 320xRGBx240 at 6bpp (18 bits/pixel), and that your display controller deals with a single large blob of data without issue. You're looking at 169 kB of data per image, and at 25 fps that's 4219 kBps or around 34 Mbit/s, which is a bit excessive. Dropping to 15 fps gives you 2531 kBps (~20 Mbit/s). Since you can just initialize your display controller and stream data at it (you don't have to read it into the micro and then re-send it to the display - just format it in your flash memory to fit the display controller), this greatly simplifies things.
SD cards should (it's been a few years since I had to deal with them) happily stream data out constantly, so you can just set it up (seek to the start of the video), and let it rip - temporarily connecting the SD card MISO pin to the display MOSI pin. If the cards spit out status data between each 512B block, just take the display chip select low.
(So essentially you just set things up, reconfigure some pins and maybe an analog mux or two, then issue 128 32-bit reads, disable CS on the display, read 1-4 status bytes, re-enable CS, and rinse & repeat.)
I'm sure there's a good few gotchas involved, but there's no reason you can't get it going. The SD card IO bus is spec'd to 25 MHz, so that's plenty - all you really have to do is generate a 25MHz SPI bus clock, then write some commands in between. Oh, and possibly twiddle the clock speed so that you get the correct playback speed. You'll also have to format the pictures properly before writing them to the card, and you have to write them contiguously, because this doesn't really care about filesystems (I'd probably build a binary image of the card and just dump it to the card using dd or something).
Also, I didn't touch on audio. Ignoring possible sync issues (Have fun!), I'd just dump it into some flash or SRAM and let a whole separate micro deal with it. You could surely get it out alongside the video, but it'd just be an exercise in frustration - and depending on your video length, audio should fit in a few MB of flash.
tl;dr - Some cleverness and a micro is all you need. You do not need a full-blown computer (RPi) or similar; it's just data.
Also, expect around 150 MB per minute of video, so a 2GB card is going to top out at around 13 minutes.