Given a hypergraph G, its hypergraphic polytope P_G is the Minkowski sum of simplices corresponding to each hyperedge of G. Using a notion of orientation on G, we prove that the faces of P_G are in bijective correspondence with acyclic orientations of G. This allows us to give a geometric understanding of the antipode in a cocommutative Hopf algebra of hypergraphs. We also give a characterization of when a hypergraphic polytope is a simple polytope. The correspondence between faces and acyclic orientations is used to prove some combinatorial properties of nestohedra and generalized Pitman-Stanley polytopes.