In this paper, the effects of viscosity contrast between the components on the rheology and morphology of phase-separating binary fluids have been studied by numerically solving time-dependent Ginzburg-Landau equation and Navier-Stokes equation. It is found that, the viscosity contrast between the components strongly affects the bulk contribution to the overall rheological behavior. When the viscosity of the minor phase is higher, the trend of bulk contribution is contrary with that of interfacial contribution. Therefore, the non-Newtonian behavior is weakened. When the viscosity of the minor phases is lower, the evolution of bulk contribution is the same as interfacial contribution, and thus the non-Newtonian behavior is strengthened. However, the interfacial contribution still plays the crucial role in the overall rheological behavior. When the system contains two droplets or many domains, due to the merging of domains induced by shear flow, there is a decrease of interfacial volume fraction. Therefore, the interfacial contribution of rheology decreases and forms a peak at small shear strain. Correspondingly, the bulk contribution has also been altered. However, whether the bulk contribution increases or decreases will depend on the viscosity contrast.