In this article, a detailed performance comparison is made between ballistic and dissipative quantum transport of metal oxide semicondutor-like graphene nanoribbon field-effect transistor, in ON and OFF-state conditions. By the self-consistent mode-space non-equilibrium Green's function approach, inter-and intraband scattering is accounted and the role of acoustic and optical phonon scattering on the performance of the devices is evaluated. We found that in this structure the dominant mechanism of scattering changes according to the ranges of voltage bias. Under large biasing conditions, the influence of optical phonon scattering becomes important. Also, the ambipolar and OFF-current are impressed by the phononassisted band-to-band tunneling and increased considerably compared to the ballistic conditions, although sub-threshold swing degrades due to optical phonon scattering.