Take a look at the Dojo Object Harness (DOH) unit test framework which is pretty much framework independent harness for JavaScript unit testing and doesn't have any Dojo dependencies. There is a very good description of it at Unit testing Web 2.0 applications using the Dojo Objective Harness.
If you want to automate the UI testing (a sore point of many developers) — check out doh.robot (temporary down. update: other link http://dojotoolkit.org/reference-guide/util/dohrobot.html ) and dijit.robotx (temporary down). The latter is designed for an acceptance testing.
Update:
Referenced articles explain how to use them, how to emulate a user interacting with your UI using mouse and/or keyboard, and how to record a testing session, so you can "play" it later automatically.
