Our understanding of gas torques on massive black hole binaries (MBHBs) relies on hydrodynamical (HD) simulations of circumbinary disks (CBDs) in which an α-viscosity models angular momentum transport. However, CBDs around MBHBs near merger are hot, magnetized plasmas where transport is driven by magnetohydrodynamical (MHD) turbulence sustained by the magnetorotational instability (MRI). We test how well an α-viscosity captures MRI turbulence with 3D global Athena++ simulations of a locally isothermal CBD around an equal-mass, circular MBHB, run in HD and MHD from the same initial torus on identical grids, so that only the transport mechanism differs. Excising the region within twice the binary separation, r<2 a_(rm bin), allows 1000 orbits of evolution, several times longer than previous MHD simulations. Both disks deliver the same accretion rate, and the gas beyond 2 a_(rm bin) exerts a positive torque in both, yet the MHD torque is 3.5 times larger and 3 times more variable. Near the midplane, transport is similar in the disk body but three times weaker in MHD at the cavity edge, where 2-3 times more mass piles up in a more eccentric cavity, raising the torque. The accretion rate is modulated mainly at twice the binary frequency in HD, but by a persistent lump at the cavity edge and its beat with the binary in MHD. The outer-disk torque therefore implies a contribution to the gas-induced phase shift in the gravitational-wave signal of LISA binaries 3.5 times larger than in α-disk models, and a different variability signature for LSST and Roman.