We report the effects of anomalous dissipation with a vanishing effective friction, which can induce ballistic diffusion and dissipative acceleration, where the fluctuation-dissipation theorem is fulfilled. An influence factor is introduced in order to describe the role of non-Markovian friction and the force-folded effect on the long-time results. The velocity-dependent coupling and force might be a dynamical origin of this dissipation. The steady acceleration of a particle moving in periodic and magnetic-force potentials are calculated.