This might be fairly easy to do.
1) Monitor the current into the projector. This is most easily done by using a clamp-type current sensor clipped over either the Line or Neutral conductor going into the projector. Have your local electrician install the sensor inside the junction box that holds the AC power receptacle that the projector plugs into.
One such device is available from Seeed Studio Non-Invasive Current Sensor 20A Note the triple "E" in the company name.
The output from the sensor is an AC current that is proportional to the current sensed. You will convert that current to a voltage for easier detection.
There are a variety of circuit types available that can detect when a voltage is above or below a certain threshold. The output of this circuit will then control the actuator that opens and closes the projector door.
The reason this is easy is that the idle current of the projector is much, much lower than when the lamp is illuminated. Open the door when the current goes above the threshold. Close the door some time period after the current drops down below the threshold.
If I were doing this project, parts count would be minimal. The current sensor, a few passive components, an 8-pin PIC microcontroller, the output relay, a tiny power supply for the PIC.
If you were really clever, you might be able to power the PIC with the signal from the current sensor.
Note that the only reason I would use a PIC microcontroller is for the timer that keeps the door open after the lamp goes out. Easier to do that timer with a few lines of code than all of the components needed for a timer or counter chip.