The dynamics of the partially ionized solar photosphere and chromosphere can be described by a set of equations that are structurally similar to the magnetohydrodynamic equations, except now the magnetic field is no longer frozen in the fluid but slips through it due to non-ideal magnetohydrodynamic effects which are manifested as Ohm, ambipolar and Hall diffusion. Macroscopic gas motions are widespread throughout the solar atmosphere and shearing motions couple to the non-ideal effects, destabilizing low-frequency fluctuations in the medium. The origin of this non-ideal magnetohydrodynamic instability lies in the collisional coupling of the neutral particles to the magnetized plasma in the presence of a sheared background flow. Unsurprisingly, the maximum growth rate and most unstable wavenumber depend on the flow gradient and ambient diffusivities. The orientation of the magnetic field, velocity shears and perturbation wavevector play a crucial role in assisting the instability. When the magnetic field and wavevector are both vertical, ambipolar and Ohm diffusion can be combined as Pedersen diffusion and cause only damping; in this case only Hall drift in tandem with shear flow drives the instability. However, for non-vertical fields and oblique wavevectors, both ambipolar diffusion and Hall drift are destabilizing. We investigate the stability of magnetic elements in the network and internetwork regions. The shear scale is not yet observationally determined, but assuming a typical shear flow gradient of ∼0.1 s-1 we show that the magnetic diffusion shear instability grows on a time-scale of 1 min. Thus, it is plausible that network-internetwork magnetic elements are subject to this fast growing, diffusive shear instability, which could play an important role in driving low-frequency turbulence in the plasma in the solar photosphere and chromosphere.