We measure, by photonic torque microscopy, the nonconservative rotational motion arising from the transverse components of the radiation pressure on optically trapped, ultrathin silicon nanowires. Unlike spherical particles, we find that nonconservative effects have a significant influence on the nanowire dynamics in the trap. We show that the extreme shape of the trapped nanowires yields a transverse component of the radiation pressure that results in an orbital rotation of the nanowire about the trap axis. We study the resulting motion as a function of optical power and nanowire length, discussing its size-scaling behavior. These shape-dependent nonconservative effects have implications for optical force calibration and optomechanics with levitated nonspherical particles.