We study the effects of the tensor force, present in modern effective nucleon-nucleon interactions, in the spin instability of nuclear and neutron matter. Stability conditions of the system against certain very low energy excitation modes are expressed in terms of Landau parameters. It is shown that in the spin case, the stability conditions are equivalent to the condition derived from the spin susceptibility, which is obtained as the zero-frequency and long-wavelength limit of the spin response function calculated in the random phase approximation. Zero-range forces of the Skyrme type and finite-range forces of M3Y and Gogny type are analyzed. It is shown that for the Skyrme forces considered, the tensor effects are sizable and tend to increase the spin instability, which appears at smaller densities than in the case that the tensor is not taken into account. On the contrary, the tensor contribution of finite-range forces to the spin susceptibility is small or negligible for both isospin channels of symmetric nuclear matter as well as for neutron matter. A comparison with the spin susceptibility provided by realistic interactions is also presented.