By making use of the method of molecular dynamics, we study the behaviors of heat conduction in one-dimensional systems under the influence of damping. The results show that when the integrability of the systems is destroyed by the damping, linear temperature profiles appear even in the harmonic lattices. Weak damping acts as a sort of perturbation in nonintegable systems and does not substantially change their temperature profiles and the thermal conductivity, while it causes essential change in integrable systems. However, in the case of strong damping, because of the excessive dissipation of energy, temperature profiles in a sunken shape and the convergent thermal conductivity appear.