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We address dynamics of open many-body systems governed by the Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) equation. We attempt to solve this equation in the Heisenberg representation, i.e. for observables, not states. We demonstrate that there are broad classes of models where the GKSL equation can be solved (essentially) exactly for certain observables. In the simplest case, the only effect of dissipation is an exponential decay on top of a coherent dynamics. This is true, in particular, for the total energy, provided the Hamiltonian is an eigenoperator of the dissipation superoperator - no matter whether the model is integrable or not. In more complex cases, dissipation alters the dynamics in a much more profound way. As an example, we solve the GKSL equation for a set of observables in a dissipative one-dimensional $XX$ model. It turns out that the observables experience the Wannier-Stark localization in the Krylov space of operators. As a result, the expectation values of the observables are linear combinations of a discrete set of decay modes.