[Update]
Microsoft released a statement clarifying the possibility of an adapter for older headsets on Xbox One. Because of the new proprietary chat port on the Xbox One controller, "Xbox plans to develop solutions in the near future to allow consumers to connect many brands of wired gaming headsets to the Wireless Controller for gaming and chat audio."
While it doesn't explicitly mention the production of a possible adapter, Microsoft's statement also doesn't deny it.
Source: Game Informer
[Original Post]
In a discussion with Aron Drayer, director of marketing for Astro gaming accessories, Game Informer sat down to find some answers about the future of headsets on consoles.
Not only will Xbox One owners not be able use their old headsets, any other accessories like Madcatz's fight sticks, will also not work on the new console. Microsoft wants to introduce new technology for the console and that means making previous technology incompatible. Astro then again confirmed that "only Xbox One controllers and accessories will work with the new console."
When speaking of the new Xbox One headsets, Drayer clarified that you can still output game audio to the headset, but players "will definitely not be able to do voice chat." The definite and explicit mention of old headsets not supporting voice chat rules out any possibility of an adapter for existing Xbox 360 headsets.
The nature of the new technology on the Xbox One controller makes an adapter impossible to produce.
Drayer likes the Microsoft's direction of the Xbox One. "Everybody is introducing new products, because the experience that Microsoft is developing on the Xbox One demands a different type of solution. We love what they are doing, and its super progressive, and we're really excited to potentially partner with them."
You can easily see how Astro benefits financially from Microsoft's proprietary chat port as it forces people to purchase all new expensive headsets once again. For already existing Astro headset owners, the chat capabilities now seems even more limited after Microsoft chose to not include a headset with every Xbox One bundle.
Source: Game Informer