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Quantum tunneling remains unexplored in many regimes of many-body quantum physics, including the effect of quantum phase transitions on tunneling dynamics. In general, the quantum phase is a statement about the ground state and has no relation to far-from-equilibrium dynamics. Although tunneling is a highly dynamical process involving many excited states, we find that the quantum phase of the Bose-Hubbard model determines phase-dependent tunneling outcomes for the quantum tunneling escape, or quasi-bound problem. Superfluid and Mott insulator correlations lead to a new quantum tunneling rate, the quantum fluctuation rate. This rate shows surprising and highly dynamical features, such as oscillatory interference between trapped and escaped atoms and a completely different macroscopic quantum tunneling behavior for superfluid and Mott insulator phases. In the superfluid phase we find that escape dynamics are wave-like and coherent, leading to interference patterns in the density with a rapid decay process which is non-exponential. Quantum entropy production peaks when about half the atoms have escaped. In the Mott phase, despite stronger repulsive interactions, tunneling is significantly slowed by the presence of a Mott gap, creating an effective extra barrier to overcome. Only one atom can tunnel at a time, yet the decay process is nearly linear, completely defying the single-particle exponential model. Moreover, quantum entropy peaks when only about one quarter of the atoms have escaped. These and many other such effects go beyond the usual notions of single-particle quantum tunneling, quantum statistical effects on tunneling, and well-known semi-classical approaches from WKB to instanton theory. These results thus open up a new regime of exploration of far-from-equilibrium dynamics for quantum simulators and quantum dynamics.