If the app you open also uses the camera or microphone, it could create a conflict, causing FaceTime to pause or stop the camera feed.
Go to Settings > General > Picture iPicture > toggle "Start PiP Automatically" off, restart the device and toggle it back on.

All third-party apps do not support PiP.
If you encounter any challenges with third-party applications, the best course of action is to directly contact the app developer for further assistance. Here's a brief guide on how to do it:
- Identify the app developer.
- Visit their website or app store listing.
- Look for their contact information or support channels.
- Clearly describe the issue you are facing.
- Provide any necessary information they may need.
- Follow their instructions and provide feedback.
- Be patient and follow up if needed.
By reaching out to the app developer, you increase your chances of receiving accurate and personalized support for the challenges you're experiencing. To contact an App developer --> How to contact an app developer - Apple Support