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Entropy production provides a general way to state the second law of thermodynamics for nonequilibrium scenarios. In open quantum system dynamics, it also serves as a useful quantifier of the degree of irreversibility. In this work we shed light on the relation between correlations, initial preparation of the system, and non-Markovianity by studying a system of two harmonic oscillators independently interacting with their local baths. Their dynamics, described by a time-local master equation, is solved to show, both numerically and analytically, that the global purity of the initial state of the system influences the behavior of the entropy production rate and that the latter depends algebraically on the entanglement that characterizes the initial state.