Okay, first here is the low down on movies/tv shows with iTunes and the iPhone.
The iTunes store sells all TV shows in SD (Standard Definition) and some shows in HD. When it comes to movies they sell/rent all movies in SD and if you have an Apple TV they also sell/rent HD versions of movies. I am not sure why HD movies are only available on the Apple TV, but I would guess it has to do with file size and the way Apple TV is always running and can continuously download very large files.
Now if you do download an HD TV show in iTunes, you will not only get the HD version for viewing on your computer, but will also get an SD version which is viewable on the iPhone. Again size and screen resolution is an issue here, a single HD movie would probably max out an 8GB iPhone so it doesn't make sense to put the HD version on it, plus the iPhone doesn't have a resolution high enough for viewing HD content.
So now you have a little background and I can answer your original question about the "Video Out" capabilities of an iPhone and your old iPod for that matter. There are currently two cables available for using the "Video Out" capabilities of the iPhone and iPod.
The Apple Composite AV Cable
The Apple Component AV Cable
Both of these cables cost $49. The composite is the older set of cables, which I am sure you have seen with a video cable (Yellow) and the two audio cables (Red and White). The component has the newer set of cables with three video cables (Red, Green and Blue) and two audio cables (Red and White). You should check on the back of your tv, you probably have one input with composite and another with component. The component is technically better because it splits out the Red, Yellow and Blue colors into their own specific cables, making for more vivid images with better colors. However, this type of cable is still considered analog and can only deliver SD resolution(480p). After reading a bit about the comparisons on these cables, it turns out the Composite being slightly earlier technology actually provides a smoother image, compared to the Component which has a crisper image, but will accentuate the flaws in the lower resolution video.
So to answer your original question yes you can buy/rent a movie from the iTunes store and using the Component or Composite cable put it on your TV. The resolution will not be HD, but it should be near/at regular DVD quality. You could also buy an HD TV show from iTunes, but unfortunately putting the movie onto the iPhone will result in a SD signal for your TV.
Finally, there is another option for you assuming the iPhone is synced with a MacBook. There are also a couple different cable/adapter you can buy for the MacBook to output to a TV. I bought a $19 cable to convert from Mini-DVI to VGA because my TV has a VGA input. However, if you don't have a VGA port (its the same connection as you use to connect a PC monitor) there is a way to convert to HDMI. If you have the white MacBook then you need a Mini-DVI -> DVI adapter ($19) AND if you the Aluminum MacBook then you need a Mini DisplayPort -> DVI adapter ($29). You then can purchase a DVI -> HDMI cable ($20).
The beauty of this set up is that you can use your TV as a monitor for the MacBook. We use it like this when doing video chats, because we can sit on the couch and chat. The other benefit is that you can take advantage of the Front Row software on the Mac which provides a full screen interface for accessing content from iTunes. http://www.apple.com/macosx/features/300.html#frontrow The MacBook actually comes with a remote which you can use to control Front Row, so you can sit on the couch and control your computer. I am not certain, but using this methodology you could also possibly watch HD TV shows purchased in the iTunes store.
If your TV supports VGA then the MacBook is the way to go, but if not then it is really a toss up.