I couldn't figure out why it was trying to do this until I realized it was because of a new feature in Lion called Resume. By default, if you quit an app, when you start it later it tries to restore the state to exactly where it was when you quit. I'd had a document on the network open when I quit Word the last time I used it, and it was trying to open that document on startup, to resume where it left off. But Word couldn't access the network and got stuck.
It is possible to turn off Resume globally, but it's probably a good feature once you get used to it, and I've learned long ago not to swim against the tide of new operating system features.
In my case, I solved the problem by starting Word in safe mode by holding the Shift key. Maybe in the future I'll be more deliberate about closing docs rather than just quitting an app. But the real solution is for Microsoft and all Mac developers to be aware of the Resume feature and be prepared to recover gracefully if something goes wrong.