During a press conference held recently, Square Enix confirmed that players that own a Final Fantasy XIV: A Realm Reborn PlayStation 3 version will be granted automatic access to PlayStation 4 Beta tests. The beta tests will start at the same time as the console’s launch on February 22nd, 2014.
Beside Beta access PS3 user will get the PS4 version of the game for free, at least in Japan. Will keep you updated if anything new arise.
UPDATE: As it was first unclear if this announcement will be applied to the NA version here is the official confirmation
“Beta Test on the new PlayStation 4 will start on February 22, 2014 and players on the current Playstation 3 system will be able to migrate to the Playstation 4 system at no additional cost, when the system is available in North America.
Players who have purchased the PlayStation 3 system version can transition their character data to the PlayStation 4 system to continue their adventure.”