There are three components necessary for a complete solution:
I don't think any of the comments here, nor the accepted answer, handle all of these scenarios.
As there's quite a bit involved, I think it's neatest as a small jQuery plugin: https://github.com/aidanlister/jquery-stickytabs
You can call the plugin like so:
$('.nav-tabs').stickyTabs();
I've made a blog post for this, http://aidanlister.com/2014/03/persisting-the-tab-state-in-bootstrap/