I was looking into whether a vanilla Android app running on a tablet can have the tablet screen output to TV via an AVI or HDMI output jack? The intent is NOT to view video content but just the app screen on a large TV. An example of how this would be useful would be a 'low-tech' training session for a bunch of people on how to use the app.
This may depend on the tablet, but as far as I know anything with Honeycomb and an HDMI output will simply mirror the screen when plugged in. My Motorola Xoom, for example, will mirror everything on the screen to a TV when my HDMI cable is plugged in. I can use it for app demos and the like at work just fine, so what you're looking for is (from a purely technical standpoint) possible.
Saying that it will work on every device would a very broad and presumably incorrect generalization on my part, though, so I would assume that I cannot accurately apply this rule to every device out there. You'll probably want to look at the specs for specific devices before you purchase one, and ask a retailer (or a manufacturer, if necessary) about it's video output capabilities.
On top of that, there are also some devices like the Galaxy Tab 7 which support composite output via a special connector cable. This also mirrors the screen, as I recall, but I don't know of any other devices that support composite hookups offhand.