Final Fantasy 14: A Realm Reborn producer Naoki Yoshida has explained that after you transfer your games from Playstation 3 to Playstation 4, you will actually lose your Playstation 3 copy of the game.
The reason for this is that the process actually moves your login information from Playstation 3 to Playstation 4. Therefore, when the transfer is complete, your login is no longer in your Playstation 3 and you will no longer be able to use it there. Of course, even if you have both your Playstation 3 and Playstation 4 in the same room, you will only be able to play Final Fantasy 14 on your Playstation 4.
To make a relatable comparison, think of the Final Fantasy 14 login like the Windows installer registration key. Microsoft has arranged Windows 7 and 8 installations in such a way that even if you borrowed an installer disc from a friend, you would still need a registration key to complete installation, otherwise Windows will not work. You can use the same installer disc to download the OS to different computers, but each one will need a different registration key.
In the same way, if you really want to be able to play Final Fantasy 14 on both your Playstation 3 and Playstation 4, you will want to buy another copy of the game. Obviously, if you wanted to keep playing your Playstation 3 game, it may be preferable to just go ahead and buy a brand new copy for the Playstation 4.
Yoshida explains the move was taken to eliminate any costs from upgrading, but this may also be a safeguard to ensure servers will not have to worry about increased logins. Fans who were told they were getting this free upgrade may not like this move, but then again, they may prefer it to having servers constantly over capacity. We will see how well this arrangement will pan out.
Source: Siliconera