Most people probably don't test static websites before deployment (except for manually going through a few pages), but it can be beneficial to test them.
What would you even test you ask?
This is especially useful when using static site generators. I wrote such tests for Fedora Developer website that is built with Jekyll.
Do you test your static websites?
If you are interested I wrote about writing such automated tests in Ruby with Capybara and Webkit.
I've never written a formal test harness for my static sites, but the very nature of static sites needing to be compiled lends itself to being a decent smoke test / indicator that things are /mostly/ all right. A jekyll build will fail from bad liquid templates, something borked in the config, etc.