It’s happened a bunch of times, and it usually goes Ok. Static sites usually handle it pretty well (with the exception of really large files - movies, etc. - getting linked to). Dynamic sites tend not to scale as well, particularly if the code is poorly written or wasn’t written with scalability in mind. If you know it’s coming, you can often do something to create a static cache of dynamic information and rebuild it periodically, depending on the content and how it needs to be presented.
Obviously, it makes DH look good to be able to handle a slashdotting gracefully, and they’re usually going to be pretty cool about it if they possibly can be. In certain circumstances, they’ve even temporarily moved a site to its own machine.
One of the worst ones I remember is when one of the Mozilla developers had his personal blog linked to by Google News, Yahoo! News, and Slashdot (and possibly the NYT), all in the same day.