Summary: We establish weak continuity results for quasilinear elliptic and subelliptic operators of divergence form, acting on corresponding classes of subharmonic functions. These results are analogous to our earlier results for fully nonlinear \(k\)-Hessian operators. From the weak continuity, we derive various potential theoretic results including capacity estimates, potential estimates and the Wiener criterion for regular boundary points. Our methods make substantial use of Harnack inequalities for solutions.