Let's take the local football game, for example. Reading the tweets will probably only whet a reader's appetite for more, especially if it was a particularly dramatic game. So invite them (via tweets with links) to come on over to the website and watch clips of the best plays. Or read in-depth analyses of what went right (or wrong). And I should note this isn't a completely original -- or unproven -- idea. As I wrote about in a previous post, the NBA had been using a similar offering to drive traffic to their website.
Similarly, live-tweeting a mayor's arraignment just drives appetite for all the Web-based goodies related to that story. After you're done with the play-by-play, shoot over a few tweets describing the related video clips, podcasts, blog posts, photos, etc... available on your website. Junkies will click on through.
One more note: To do this successfully, newspapers are going to have to set up tweets differently than they do now. Most of the ones that are tweeting offer a single tweet channel that includes everything from the latest crime news, to football player trades, to city council goings on. Again, this is an implementation by someone who thought Twitter was the same as RSS.
Instead, smart news organizations will set up seperate Twitter channels for separate types of news or events. There would have to be some kind of mechanism for letting readers know what kind of channels were coming up (eg: "Sign up for tomorrow's tweets on the Kilpatrick arraignment" or "Follow the Cougars-Buccaneers game on Twitter on Thursday"). But the point is: People will want to follow discrete events, rather than getting ongoing grab bags of everything, so this is something news organizations will have to figure out how to set up.