We suggest a quantum simulator that allows to study the role of memory effects in the dynamics of open quantum systems. A particular feature of our simulator is the ability to engineer both Markovian and non-Markovian dynamics by means of quantum measurements and the quantum Zeno dynamics induced by them. The simulator is realized by two subsystems of a bipartite quantum system or two subspaces of a single system which can be identified as system and meter. Exploiting the analogy between dissipation and quantum measurements, the interaction between system and meter gives rise to quantum Zeno dynamics, and the dissipation strength experienced by the system can be tuned by changing the parameters of the measurement, i.e., the interaction with the meter. Our proposal can readily be realized with existing experimental technology, such as cavity- or circuit-QED platforms or ultracold atoms.