In this article we present a formula for the plurigenera of minimal models of nondegenerate toric hypersurfaces, which is valid in any dimension and which expresses these invariants through lattice points on the Fine interior. Besides we consider other birational models of toric hypersurfaces and study their singularities from the point of view of the minimal model program. We show that the first irregularity q(Y ) of a minimal model of a toric hypersurface is always 0. Restricting to surfaces in toric 3-folds we compute the difference in the rank of the Picard group when switching between birational models.