The flight instructor reported that he and two student pilots were on a night, visual flight rules, instructional flight. The instructor tried to activate the pilot-controlled lighting at the destination airport, but he believed that it was inoperative. He reported that he could see the wind sock on the airfield but that he did not see the "X" near the runway numbers and performed a touch and go. During rotation the instructor reported that, "I heard a red cone make impact with the nose gear section." He had to apply continuous forward pressure to the yoke because the nose continued to pitch up with the trim set to the full-down position. He asserted that the flight characteristics were "acceptable" and continued the flight about 47 nautical miles to their home airport. Upon arrival, the instructor alerted the tower that he had a stabilator malfunction and landed the airplane with zero flaps. The instructor reported that he did not check the notices to airmen (NOTAM). The airplane sustained substantial damage to the stabilator. The pilot reported that there were no preaccident mechanical malfunctions or failures with the airplane that would have precluded normal operation. According to Federal Aviation Administration NOTAM 03/058, the airport runways were closed at the time of the accident.