For some reason after updating to iOS 10 beta 2, users were signed out of their account for security reasons, but unfortunately attempting to sign back in was not allowed and they were also prevented by resetting their passwords. They attempted to reset their password from Apple’s iForgot page but were given the error messaging saying, “Your request could not be completed at this time,” despite the fact that there were no issues with the service.
It has been speculated that this could be due to two-factor authentication. According to the majority of the posts from users with the problem, they have seem to have two-factor turned on which is why some speculate that two-factor could be the problem behind this issue. Unfortunately there isn’t much one can do at this point except request for an account recovery.
It is unclear if this is an issue that Apple has addressed prior to the release of the public beta of iOS 10, but maybe if you want to play it safe, perhaps you should hold off on getting the public beta for now.