The pilot reported that during the landing roll out a vibration was observed coming from the nose wheel. As the airplane was decelerating to a stop, smoke was noticed coming from the nose gear area. A postaccident examination of the airplane revealed that during the landing roll out the nose wheel tire had gone flat. This resulted in the vibration of the nose landing gear, and the subsequent fire to the tire and forward section of the fuselage; the fuselage sustained substantial damage as a result of the thermal activity. The examination also revealed no anomalies with any of the nose landing gear components that would have precluded normal operation.