Has this also. Easily fixed but pretty scary.
1) Download the latest iOS version for your device from the downloads page.
2) Unpack the archive.
3) Ensure you have updated to the latest version of iTunes
4) Connect your iphone (which is likely saying iOS has expired, it won't activate or allow you to backup to iTunes or iCloud)
5) Control click or command click (depending if Windows or Mac) to open the iOS 9 Beta 5 image IPSW file from the "Update" button.
6) Read and follow all onscreen prompts, note they say "Update" and NOT "Restore". If they do say "Restore", stop, you have gone wrong.
My phone rebooted, went into what looked like recovery, sat there for five minutes and looked as if it hung, then rebooted. The computer said it timed out but the phone then offered "Swipe to upgrade", I did this and was back up and running. 30 minute procecure (excluding downloading image file)
7) Take a fresh backup and confirm iCloud backups are still functioning (mine were not).
I didn't loose any data or settings - thankfully! It was just like an over air patch update rather than a full install, something I didn't realise could be done via iTunes. I was doing all of this from a Windows laptop. I went from Public Beta 1 to Developer Beta 5 on a iPhone 6 Plus.