An electronic device such as a head-mounted device may have gaze tracking sensors and other circuits that include packaged light-emitting devices. The packaged light-emitting devices may each include a light-emitting component such as a light-emitting diode or laser that emits infrared light or light of other wavelengths. The packaged light-emitting devices may have protection circuitry. The protection circuitry of each device may include one or more sensors such as a package integrity sensor, a proximity sensor, a photodetector for measuring emitted light intensity, a temperature sensor for measuring a temperature associated with the light-emitting component, and a current sensor. A control circuit such as an integrated circuit containing a comparator or other analog circuitry and/or a digital control circuit may gather information from one or more of the sensors and may control the light-emitting component based on the sensor information to prevent elevated light emission intensities.