I would think that you would need at least one input device, most likely a mouse. Certainly at some point you might want to use the rewind or pause (pee break) functions. Even if you could rig it like you described, all you would be able to do is insert a DVD or eject the DVD and that's not a whole lot of options, right there.
Also, even though the DVD may auto-start, it will only take you to a main menu that gives you typical DVD options like cast biographies, making of, bonus music tracks and videos, etc. You would still need to select "Play movie" as an option. Every settop and PC DVD system I've ever seen works this way when playing DVDs. Also, some Warner Bros. DVD releases have this PC Friendly program that is basically crap and advertising which autostarts when a DVD in inserted. The option has to be denied when it prompts to install the program onto your hard drive.
The other thing is--I'm not sure what kind of TV you have, but if it is a smaller model without a lot of jacks, audio may be a bit of a challenge, unless it's a newer TV. You can buy a stereo-to-RCA jack (Radio Shack has 'em) and plug one end into your sound card and the other into the back of a stereo receiver, VCR, etc, but you might not have the plug(s) (audio in) on your TV itself.
I don't know if you have a video card already, but I read (Anandtech) that Nvidia cards with TV-out don't handle the task very well compared to Matrox or ATI cards. That's something I haven't had any experience with personally, though. I seriously doubt you would want to purchase a video card for the express purpose of playing DVDs on a television set. If that were the case, I would recommend buying a cheapy settop DVD player from Walmart, K-Mart, or Target instead and save money for a Geforce4 4400 to put (to good use) in a gaming computer.
I'm curious now--why not just connect it to a monitor and get far better resolution and color? Also, the CPU and the hard drive that you would have to use to run the OS, it seems to me, could be better used in an application, file, or print server for your network? I'm having a hard time trying to accept that an entire computer would be turned into, essentially, a single DVD player with only an on/off button.