This issue is caused by a few things playing together.
1. If you plug in or activate a new audio device such as earphones or a headset, it makes itself the "default" audio device of your PC/Laptop.
2. Common Bluetooth devices have 2 operation modes. One "high quality" mode meant for listening to music, and one voip mode for voice communication. The headset is in high quality mode as default, and switches to voip mode as soon as some app is activating the microphone of the Bluetooth device.
3. The high quality mode has one Windows device, and the voip mode has another Windows device (the name often contains text like "hands free").
4. If the mic is activated, thus voip mode gets activated, the high quality device becomes silent and audio is instead coming out of the hands free device.
5. If OBS is started, it activates every audio device in Settings->Audio. If there is a device called "default", it activates the current default audio mic device, which is the Bluetooth mic if a Bluetooth headset is connected.
6. Activating the mic makes the Bluetooth switch to voip mode, thus the high quality device becomes silent, and this is what you described.
How to fix:
If you don't need the Bluetooth mic while you use OBS, go to Settings->Audio and make sure every Mic/Auxiliary audio is either set to Disabled or to some non-Bluetooth mic device. So OBS will not initialize the mic, so the Headset will not switch to voice mode, so the high quality device will not become silent.
If you need the Bluetooth mic, go to Settings->Audio and explicitly set the hands free device as Desktop audio to capture. Change every entry that has "Default" as value and explicitly choose the hand free device. Be aware the audio quality in this mode is much less than the default audio quality. This all is a Bluetooth limitation, not a OBS limitation.