We consider a class of CR manifolds which are defined as asymptotically Heisenberg, and for these we give a notion of mass. From the solvability of the $\Box_b$ equation in a certain functional class ([Hsiao-Yung]), we prove positivity of the mass under the condition that the Webster curvature is positive and that the manifold is embeddable. We apply this result to the Yamabe problem for compact CR manifolds, assuming positivity of the Webster class and non-negativity of the Paneitz operator. This is joint work with J.H.Cheng and P.Yang.