You might want to hold off on trying the iOS 10 beta if you can't bear to be without two-factor sign-ins. At least a handful of users are reporting a two-part problem that has locked them out of their Apple IDs. The first part is a bug with app-specific passwords that can force a password reset, regardless of the iOS version you use. It's not the worst issue if you're using a stable version of iOS, since it takes mere moments to get a new password. Things may go haywire if you're an iOS beta tester with two-factor authentication turned on, however. Users say that Apple's iForgot password system doesn't work for those experimenters, shutting them out of their accounts the moment the app-specific password glitch creeps up.
Those affected by the problems say that engineers are aware of what's going on, so you might be in the clear by the time you read this. We've reached out to Apple for more details, and we'll let you know if it has more to add. One thing's for sure: this is a not-so-friendly reminder of the risks of trying pre-release operating systems with accounts and devices you can't afford to lose.
Update: Some users report that you can use a link in your two-factor authentication email notice to disable the security measure and thus recover your account. You'll still have to wait until the underlying issues are resolved to turn two-factor back on, but it beats a total lockout. Thanks, Hemmik!