Try turning iMessage off and on again in Settings - Messages.
If that doesn't help, force start your iPhone: press and hold Home and Power button at the same time, until you see the Apple logo. As soon as it appears, let go of both buttons. Your device will power up on it's own.
Also try resetting your network settings (Settings - General - Reset - Reset Network Settings). You'll have to set them up again afterwards.
If you have an iTunes backup from the time before iOS 10, restore from that backup, and try upgrading to the Public Beta version again.
If that doesn't solve your problem, send feedback to Apple, let them know about your problem, so they can work at it.