Starting from the kinetic theory description of massive spin-1/2 particles in the presence of a magnetic field, equations for relativistic dissipative nonresistive magnetohydrodynamics are obtained in the small polarization limit. We use a relaxation-time approximation for the collision kernel in the relativistic Boltzmann equation and calculate nonequilibrium corrections to the phase-space distribution function of spin-polarizable particles. We demonstrate that our framework naturally leads to emergence of the well-known Einstein-de Haas and Barnett effects. We obtain multiple transport coefficients and show, for the first time, that the coupling between spin and magnetic field appear at gradient order in the hydrodynamic equation.