We consider a class of evolutionary variational inequalities depending on a parameter, the so-called viscosity. We recall existence and uniqueness results, both in the viscous and inviscid case. Then we prove that the solution of the inequality involving viscosity converges to the solution of the corresponding inviscid problem as the viscosity converges to zero. Finally, we apply these abstract results in the study of two antiplane quasistatic frictional contact problems with viscoelastic and elastic materials, respectively. For each of the problems we prove the existence of a unique weak solution; we also provide convergence results, together with their mechanical interpretation.