No game audio unless headphones plugged in

You can plug headphones straight into the chat adaptor to receive game audio or sound from anything plugged into the xbox

The chat adapter pulls game audio and chat to the headphons. I'm using a pair of Bose headphones with an inline mic. Works a charm with headset adapter. No wire to console. Its wireless. Just plug the 4pole 3.5mm plug into adapter, and I get game audio and
chat. If you dont believe me, do some research.

Not headphones, the controller plug is smaller and used for chat, headphones won't fit. My 360 Turtle beach headset gets its game audio from the TV too.

The red & white plugs that come with the headset need to be connected to the xbox audio cables, this should solve the problem. Another possiblity is to plug the green headphone plug on the end of the headset into the TV's headphone jack.

Can you try plugging the Razer into a "normal" audio device like a MP3 player or your smartphone and play music through the headset?

Does that work?

If that works, try to plug the headset into a different controller if you have one.

In the console settings, go to "Display & Sound" and in the audio settings, make sure that audio output through headphones is selected. Also make sure, that you have selected "Stereo" or "Windows Sonic" as sound output, unless you have actually paid money
in the Dolby Atmos app.

If you just downloaded the Dolby Atmos App, and didn't pay them for Dolby Atmos for Headphones, the audio output will stop working after 14 days.

You can then either purchase Dolby Atmos for Headphones, or switch to Windows Sonic, which does basically the same: giving you spatial 3D surround sound via headphones. For free.