We experimentally and theoretically investigate the influence of the dipole-dipole interactions (DDIs) on the total interspecies interaction in an erbium-dysprosium mixture. By rotating the dipole orientation we are able to tune the effect of the long-range and anisotropic DDI, and therefore the in-trap displacements of the erbium and dysprosium clouds. We present a theoretical description for our binary system based on an extended Gross-Pitaevskii theory, including the single-species beyond mean-field terms, and we predict a lower and an upper bound for the interspecies scattering length . Our work is a step towards the investigation of the experimentally unexplored dipolar miscibility-immiscibility phase diagram and the realization of quantum droplets and supersolid states with heteronuclear dipolar mixtures.