The introduction of ``twist″ or relative rotation between two atomically thin van der Waals membranes gives rise to periodic moire potential, leading to a substantial alteration of the band structure of the planar assembly. While most of the recent experiments primarily focus on the electronic-band hybridization by probing in-plane transport properties, here we report out-of-plane thermoelectric measurements across the van der Waals gap in twisted bilayer graphene, which exhibits an interplay of twist-dependent interlayer electronic and phononic hybridization. We show that at large twist angles, the thermopower is entirely driven by a novel phonon-drag effect at subnanometer scale, while the electronic component of the thermopower is recovered only when the misorientation between the layers is reduced to < 6 degrees. Our experiment shows that cross-plane thermoelectricity at low angles is exceptionally sensitive to the nature of band dispersion and may provide fundamental insights into the coherence of electronic states in twisted bilayer graphene.