Does using a professional camera (DSLR/Mirrorless) damage the sensor over time and lower the lifetime of your camera? Usually shutter count is a good indicator of health of a used camera - would this not have a similarly huge impact?
I don't think there's any risk of damaging the sensor. The shutter is a mechanical part that will physically wear out over time, the sensor is converting photons into voltage.
I've been using my camera as a webcam for over 2 years without any issue, and it's usually on for several hours per day. As long as the sensor isn't getting very hot then I don't think there's much risk to the camera.