Here's the code:
Saving it into a bookmark is easy. In most browsers, you can just highlight the code and then drag and drop it onto the bookmarks bar. (You can rename it once the bookmark is saved.)
When you run this function, the Moments tab will be removed from your Twitter header. If you refresh the page, log out of Twitter and log back in, or click on certain Twitter links (such as Lists) the Moments tab will reappear. If that happens, simply click the bookmark again to remove it. As long as you stay on the "main" Twitter page -- including searching Twitter, sending or reading tweets, adding followers, and reading direct messages -- the Moments tab will remain gone.
UPDATE: This code is now available (in altered form) as the Chrome browser extension Mention It. This extension removes the Moments tab from Twitter, and replaces it with a Mentions tab. The extension runs automatically, so you don't have to click on it like you would with the bookmarklet described above.