Given a graph G on n vertices, Postnikov defined a graph associahedron  P_G as an example of a generalized permutohedron, a polytope whose  normal fan coarsens the braid arrangement. Combinatorially, each face of  P_G corresponds to certain collections of compatible subgraphs of G called tubings. Graph associahedra were introduced independently by Carr  and Devadoss and by Davis, Januszkiewicz, and Scott. In this talk, we  consider the poset obtained by orienting the one-skeleton of P_G  according to a certain linear functional and its relationship to the weak order on S_n.