This is the way Bluetooth devices work. Your headset is probably bluetooth-based. Bluetooth devices have 2 operation modes: high quality audio (output only) and communication audio (output+mic). Generally, the high quality mode is active. As soon as some app opens the mic device, the headset switches operation mode to communication audio. This causes the high quality audio device go silent and instead the communication audio playback (headset) and recording (mic) devices are activated and input/output takes place here instead.
Now, if you start OBS, it opens all devices you find in Settings->Audio. If you find a "default" here, the default Windows device is substituted here. In your case, it is probably your mic. That means, OBS activates your mic on startup, then your headset switches to communication audio mode, thus the output device changes.
To avoid this activation on OBS startup: In settings->Audio set the Mic devices to disabled or change the "default" settings to some device that is not your headset.
You can also do this: After you started OBS and the communication audio mode was activated, go to into the Windows sound settings and set the now active communication audio device as default device. Output from other apps will now again output to your headset.