In standard fluid dynamics, the density change associated with flow is often assumed to be negligible, implying that the fluid is incompressible. For example, this has been established for simple shear flows, where no pressure change is associated with flow: there is no volume deformation due to viscous stress and inertial effects can be neglected. Accordingly, any flow-induced instabilities (such as cavitation) are unexpected for simple shear flows. Here we demonstrate that the incompressibility condition can be violated even for simple shear flows, by taking into account the coupling between the flow and density fluctuations, which arises owing to the density dependence of the viscosity. We show that a liquid can become mechanically unstable above a critical shear rate that is given by the inverse of the derivative of viscosity with respect to pressure. Our model predicts that, for very viscous liquids, this shear-induced instability should occur at moderate shear rates that are experimentally accessible. Our results explain the unusual shear-induced instability observed in viscous lubricants, and may illuminate other poorly understood phenomena associated with mechanical instability of liquids at low Reynolds number; for example, shear-induced cavitation and bubble growth, and shear-banding of very viscous liquids such as metallic glasses and the Earth's mantle.