We consider the question of the existence of a generalized H-theorem in the context of the variational method in the information-theoretical approach that generates the nonequilibrium statistical operator formalism. After briefly reviewing how the latter provides mechano-statistical foundations for phenomenological irreversible thermodynamics, the so-called Informational Statistical Thermodynamics, we discuss how dissipative phenomena are accounted for by the procedure. Such effects are related to a generalized H-theorem and a weak criterion of positive entropy production. These results are a consequence of the definition of a coarse-grained statistical entropy, resulting from the projection of the full nonequilibrium distribution on the subspace of the slow relaxing dynamical quantities, that are appropriate for the description of the irreversible evolution of the system from the state of initial preparation.