PA-28 instructor pilot with trainee reported an engine fire during engine run-up prior to takeoff.

Narrative

The training event started at XA:00 on Day 0. The student pilot performed a routine preflight inspection on the airplane. I double checked his work; and no abnormalities were noted by him or myself. We planned to stay in the traffic pattern at ZZZ. Engine start; after start and taxi procedures had no abnormalities and all engine indications were within their respectful ranges. We taxied to runway XX via taxiways 1 and 2 and performed a run-up as per the checklist. When we tested the right magneto by grounding the left; the RPM drop was outside of its tolerance of 175 RPM; dropping by at least 200 RPM. I believed it was caused by a carbon buildup on the spark plugs (as this is usually the case; and there was a student pilot who had gone on a solo cross country just before our flight who may have run the mixture too rich). I increased power to full and leaned the mixture to try to clear the carbon and we smelled what we later found out was oil burning. I then noticed dark smoke filling up the airplane and more flowing out of the nose bowl of the cowling. I immediately cut off the mixture and magnetos and notified ground control of what had happened and that we were evacuating. My student and I exited the airplane and three ARFF (Airport Rescue and Firefighting) vehicles and one Airport Operations vehicle met us at the airplane; by which time the smoking had stopped. We explained what had happened and ARFF instructed (and helped) me remove the top cowling. The fiberglass melted slightly where the heat came from. They pointed out where it had burned; monitored the temperature and sprayed water on it. They asked if we had someone who could tow us or if we preferred to wait for their tug; and I opted to have someone from the school bring the golf cart. The airplane was towed back to the flight school's ramp. I shook the fireman's hand and told him we did not require any more assistance at that time. I spoke with my bosses over the phone; then my student and I debriefed on the situation and the importance of always being ready for any emergencies that could happen in any phase of flight.The whole event as described happened in the span of just under two hours.