The PlayStation 4 will not require an internet connection, though it is likely that if you want to take advantage of some of its core features and frequent updates you'll need a connection at some point. According to Eurogamer Sony says that, while you may want to keep your console connected to the internet to take advantage of the system's new features, it is not a requirement.
Speaking to Sony Worldwide Studios boss Shuhei Yoshida after the PS4 press conference last night, Eurogamer asked about having to be connected to the Internet in order to use the new console. Here's the exchange on that:
Eurogamer: Does the PlayStation 4 always need to be connected to the internet?
Yoshida: You can play offline, but you may want to keep it connected. The system has the low-power mode - I don't know the official term - that the main system is shut down but the subsystem is awake. Downloading or updating or you can wake it up using either the tablet, smartphone or PS Vita.
When asked if all the features Yoshida mentioned in his previous answer were optional and customizable, he said:
"Oh yes, yes, you can go offline totally. Social is big for us, but we understand there are some people who are anti-social! So if you don't want to connect to anyone else, you can do that," he responded.
Of course, if you want to play games like Diablo III or Bungie's Destiny then you already should know that a connection is going to be required - and in the case of Diablo III - that requirement also applies to single-player.