Hello maren229,
If your FaceTime photos are not saving, you can try resetting the option. Go to Settings > FaceTime and turn off the FaceTime Live Photos option. After disabling that, restart the device.
Once the device has restarted, go back to Settings > FaceTime and re-enable the FaceTime Live Photos option. After that, test it out to see if it works.
Take a Live Photo in FaceTime on iPhone
Also, make sure the software on your device is up to date.
Update your iPhone
If the photos still do not save, you'll want to contact Apple to report the issue: Get Support
Cheers!