Sony has confirmed to Game Informer that the PlayStation 4 will not recognize any bluetooth headsets, except the PULSE Gaming headset, for chat.
The platform holder confirmed that if you own a Sony-branded PULSE gaming headset (or Elite edition) you will need to wait for a system update to use these headsets. They will not work at launch. Any other headset that relies on bluetooth to connect will not work at all.
Headsets that rely on a USB connection for chat will work, but not at launch. A future system update will add compatibility for these headsets. Finally any headset that relies on digital optical for game audio will work out of the box.
The situation is similar to the Xbox One, which will require an adapter if you wish to use Xbox 360 headsets. Both the Xbox One and PlayStation 4 will come with a headset in the box.
Source: Game Informer