We show that weak measurement leads to unconventional quantum Zeno dynamics with Raman-like transitions via virtual states outside the Zeno subspace. We extend this concept into the realm of non-Hermitian dynamics by showing that the stochastic competition between measurement and a system's own dynamics can be described by a non-Hermitian Hamiltonian. We obtain a solution for ultracold bosons in a lattice and show that a dark state of tunneling is achieved as a steady state in which the observable's fluctuations are zero and tunneling is suppressed by destructive matter-wave interference.