  We investigate the time evolution of an open quantum system described by a
Lindblad master equation with dissipation acting only on a part of the degrees
of freedom ${\cal H}_0$ of the system, and targeting a unique dark state in
${\cal H}_0$. We show that, in the Zeno limit of large dissipation, the density
matrix of the system traced over the dissipative subspace ${\cal H}_0$, evolves
according to another Lindblad dynamics, with renormalized effective Hamiltonian
and weak effective dissipation. This behavior is explicitly checked in the case
of Heisenberg spin chains with one or both boundary spins strongly coupled to a
magnetic reservoir. Moreover, the populations of the eigenstates of the
renormalized effective Hamiltonian evolve in time according to a classical
Markov dynamics. As a direct application of this result, we propose a
computationally-efficient exact method to evaluate the nonequilibrium steady
state of a general system in the limit of strong dissipation.
