The dissipative dynamic stability is investigated of dark solitons in elongated Bose-Einstein condensates that can be described by the Gross-Pitaevskii equation including an additional term. Based on the direct perturbation theory for the nonlinear Schrodinger equation, the dependence of the soliton velocity on time is explicitly given, and the shape of dark solitons remaining unchanged under the dissipative condition is confirmed theoretically for the first time. It is found that the dynamically stable dark solitons tu out to be thermodynamically unstable.