After a security hole was discovered in Apple's iForgot password recovery page that allowed passwords to be recovered with just an e-mail and date of birth, Apple has disabled the page for obvious reasons. Navigating to the page now returns a "Currently Unavailable" status message that suggests you "check back later."
Make sure to enable two-step authentication on your account to help protect from future vulnerabilities such as this.
At the time of writing, Apple's page continually timed out for both my coworkers and myself.