Using Qunit for testing JS
You can use Qunit and Sinon for automated testing of your JS front-end code using jQuery and jQuery UI (the JS libraries we use for new UI development). There are some sample tests in the views/dev/test/ directory. The Qunit and Sinon code itself lives in the htdocs/js/tests/ directory. There's some additional Dreamwidth-specific glue in DW::Controller::Dev.
For automated testing of back-end components, see Dev Testing.