We analyze the behavior of an active tracer particle embedded in a thermal lattice-gas with exclusion up to the third nearest neighbor on the square lattice. For the case with no orientational diffusion of the tracer, we derive an analytical expression for the resulting drift velocity v of the tracer in terms of non-equilibrium density correlations involving the tracer particle and its neighbors, which we verify using numerical simulations. We show that the properties of the passive system alone do not adequately describe even this simple system of a single active tracer, except at low activity. For large activity and low density, we develop an approximation for v based on equilibrium correlations of the passive system. For the case where the tracer exhibits orientational diffusion independent of its neighbors, we relate its diffusion coefficient to the thermal diffusion coefficient and v. Finally we numerically study the effect of rotational locking, where the rotation of the tracer is limited by the presence of neighboring particles.