On May 19, 2016, about 1320 eastern daylight time, a privately owned and operated Piper PA-28-180, N7781W, was substantially damaged during landing at Genesee County Airport (GVQ), Batavia, New York. The student pilot was not injured. Visual meteorological conditions prevailed at the time and no flight plan was filed for the instructional flight that was conducted under the provisions of 14 Code of Federal Regulations Part 91. The flight originated about 1300 from Akron Airport (9G3), Akron, New York.The student pilot stated that he departed from 9G3 and flew to GVQ where he performed two uneventful touch-and-go landings on runway 28. During a third touch-and-go landing he landed uneventfully, retracted the flaps, and added engine power. He was utilizing some right rudder input as the airplane reached rotation speed (approximately 70 mph), and "pulled hard to the left," which he could not correct with full application of right rudder. As the airplane approached the left side of the runway, he noted runway signage, but because the airplane was at takeoff speed, he thought climbing above the signs would result in a safer outcome and continued the takeoff. The airplane departed the left side of the runway and while attempting to clear the signage off the left side of the runway, both wings collided with separate signs, which breached both fuel tanks. The pilot continued the takeoff and while turning left to land on a nearby road, he noted fuel streaming from the left fuel tank and the engine lost power. He flew under one set of powerlines and then over a second set before landing uneventfully on the road. Examination of the airport and airplane by a Federal Aviation Administration inspector revealed browning of grass beyond both impacted airport signs, and for a short distance past the point where the airplane became airborne. Both fuel tanks were breached; neither contained any fuel. Flight control continuity was confirmed for all flight controls. Examination of the impacted airport signs revealed their posts were frangible. The pilot reported the wind to be from north at 8 mph with no gusts, while a weather observation taken about 26 minutes before the accident at an airport located about 22 nautical miles east-northeast from GVQ reported the wind was from 280° at 9 knots.