A search for dark matter is conducted in events with large missing transverse momentum and a hadronically decaying, Lorentz-boosted top quark. This study is performed using proton-proton collisions at a center-of-mass energy of 13 TeV, in data recorded by the CMS detector in 2016 at the LHC, corresponding to an integrated luminosity of 36 fb$^{-1}$. New substructure techniques, including the novel use of energy correlation functions, are utilized to identify the decay products of the top quark. With no significant deviations observed from predictions of the standard model, limits are placed on the production of new heavy bosons coupling to dark matter particles. For a scenario with purely vector-like or purely axial-vector-like flavor changing neutral currents, mediator masses between 0.20 and 1.75 TeV are excluded at 95% confidence level, given a sufficiently small dark matter mass. Scalar resonances decaying into a top quark and a dark matter fermion are excluded for masses below 3.4 TeV, assuming a dark matter mass of 100 GeV.