This paper is a sequel to  and . We continue our study of occupation time large deviation probabilities for some simple infinite particle systems by analysing the so-called voter model ζt (see e.g.,  or ). In keeping with our previous results, we show that the large deviations are "classical" in high dimensions (d≧5 for ζt) but "fat" in low dimensions (d≦4). Interaction distinguishes the voter model from the independent particle systems of  and , and consequently exact computations no longer seem feasible. Instead, we derive upper and lower bounds which capture the asymptotic decay rate of the large deviation tails.