A significant viscosity variation with the shear rate has been observed for several ionic liquids in rheometry experiments above a critical shear rate. Depending on the liquid and the rheological conditions, both viscosity increase and decrease have been reported. So far, these variations have been interpreted as a signature of a non-Newtonian behavior. However, the measured critical shear rates are orders of magnitude below the ones predicted by numerical simulations. In this work, we perform new rheometry experiments with both ionic liquids and Newtonian liquids to elucidate this discrepancy. For these two types of liquids, both a viscosity decrease and increase have been measured depending on the geometry of the rheometer and the zero-shear viscosity of the liquid. We interpret the viscosity decrease as resulting from viscous heating, since the viscosity of the investigated liquids is also highly temperature-dependent, and the viscosity increase as resulting from the development of instabilities at high shear rates.