We have calculated the viscosities of a nematic liquid crystal phase of the Gay-Berne fluid [J. G. Gay and B. J. Berne, J. Chem, Phys. 74, 3316 (1981)] by using nonequilibrium molecular dynamics simulation methods. The calculations are facilitated by applying a Gaussian constraint method that makes it possible to fix the orientation of the director. The viscosity is a fourth rank tensor, In an isotropic fluid it has got three independent components whereas it has got seven components in an axially symmetric liquid crystal. Our estimates of the shear viscosities and the twist viscosities agree with the equilibrium fluctuation results of a previous study [S. Sarman and D. J. Evans, J. Chem, Phys. 99, 9021 (1993)]. We have also found that the streaming angular velocity is different from zero even though the angular velocity of the director is constrained to be zero thus demonstrating that these two angular velocities are different quantities. Finally we have calculated the irreversible entropy production due to the symmetric traceless strain rate as a function of the alignment angle, We have found it to be minimal near the preferred alignment angle. This is in agreement with the principle of minimum entropy production of linear irreversible thermodynamics.