Signed Out when started game

I have the same problem but with call of duty BO2

Perhaps it's the beta itself. The Destiny beta ended last night so no one can play it anymore.

Are you having the same issue with other games or apps?

Hello There, In order to fix this issue you will need to delete your profile only (Not With Items) from your xbox, afterwards clear your system cache, I usually do this 3 times to make sure it's fully cleared. Then lastly, re download your profile. Enjoy
Xbox Live Let me know if you require further assistance.

TO CLEAR THE SYSTEM CACHE

Press the Guide button on your controller, go to Settings, and select System Settings.

Select Storage or Memory.

Highlight any storage device, and then press Y on your controller. (It doesn't matter which storage device you select; the cache will be cleared for all storage devices.)

Select Clear System Cache.

When prompted to confirm storage device maintenance, select Yes.

TO DELETE A PROFILE

Go to settings and select System.

Select Storage.

If you don't connect any external storage devices to your console, select Hard Drive. If you do use external storage, instead select All Devices.

Select the profile you wish to delete (Remember to delete profile only)

Have you got a strict nat setting that's maybe why your getting signed out ?

Have you tried resetting the Xbox?