I have had limited success with 2 approaches:
1) use rewire to replace the database driver library like mongodb with a mocked one, perhaps using the spy/stub/mock capabilities of sinon
2) Set your db as an app setting via app.set('mongodb', connectedDb)
for dev/prod but in test environment set a mock database instead. This requires your db-accessing code (models typically) to get the DB from the app, or otherwise be mock-friendly or designed with a dependency injection pattern.
Neither of these make everything clean and painless, but I have gotten some utility out of them.