I had the same problem, found this on Loadstone if you've not seen:

We have confirmed an issue wherein North American users who have downloaded the FINAL FANTASY XIV: A Realm Reborn (Beta Version) on the PlayStation®4 between Tuesday, April 1, and Thursday, April 3, at 4:00 p.m. (PDT) will be unable to properly log into the game.

If you have downloaded the Beta during the aforementioned time, you will need to return to PlayStation®Store and make your purchase again to resolve the issue. You will not need to re-download the software.

Here are the steps you will need to take to be able to correct the issue and participate in the Beta Test.

1. Log into the Sony Entertainment Network account that you used to download the Beta Test software.
2. On the PlayStation 4 home screen, navigate to PlayStation®Store and launch it.
3. On the menu, select “Games” > “New” > “New This Month” and navigate to the “FINAL FANTASY XIV: A Realm Reborn” content and select it.
4. Select “Add to Cart” and “Proceed to Checkout” and finally “Confirm Purchase”.

To confirm the issue has been resolved, simply navigate to the “Settings” menu on the PlayStation®4 home screen, select PSN℠ > Account Information > Services List > and select “FINAL FANTASY XIV: A Realm Reborn” where it should display “Open Beta 2 Entitlement” if the issue has been resolved.

The issue has been addressed for those downloading the Beta Test software from PlayStation®Store going forward.