Playing online multiplayer on the PlayStation 4 is going to require players to have a PlayStation Plus subscription. The policy is similar to the what Microsoft uses for Xbox 360 and Xbox One for Xbox Live Gold.
"PS4 multiplayer online access requires PSN account & PS Plus subscription," fine print says in the bottom of this Sony-released video (via Polygon). A PS+ membership runs $49.99 a year, or $17.99 if you can choose to pay for three months. Xbox Live Gold costs $59.99 a year.
You won't need a new membership for the PS4 if you already have one for Vita or PlayStation 3.