A press release from Turtle Beach and a report from Game Informer state that voice chat functionality on current-generation headsets won’t work on the Xbox One and the Playstation 4 at launch.
Since the new Xbox One controller uses a new proprietary port for chat headsets, an adapter will be needed for current-generation headsets to work with it. An adapter which, according to Turtle Beach, won’t be available until early 2014.
Sony’s issues are a bit more complicated. Users with Sony-branded headsets or PULSE gaming headsets will need to wait for a post-launch system update while headsets that rely on Bluetooth will not work at all. USB and Toslink (optical) headsets will also need to wait for a system update before you’ll be able to use them for chat.
Both the Xbox One and the Playstation 4 will come bundled with headsets so you’ll at least have something to use while you wait for either the Xbox One adapter or the system update for the Playstation 4.