We investigate theoretically the drainage of a thin liquid film confined between two hydrophobic spheres. Such a problem has been considered in Vinogradova's seminal work, emphasizing the role of slippage. However, it does not include the boundary curvature effects, which may become especially important when the slip lengths are comparable to the sphere radii. We present a reformulation of the hydrodynamic boundary conditions, with the boundary curvature effects fully taken into account. It is shown that such effects not only renormalize the effective slip lengths but also give new contributions to the pressure and drag force, neglected so far. We focus on the symmetric case of two identical spheres with the same radii and slip lengths and obtain analytical expressions for the pressure as well as the drag force. The boundary curvature corrections to the drag force are analyzed and found to be more important for more hydrophobic spheres. The implications of our results are discussed for the coagulation processes of colloids and measurements of surface forces or slip lengths with the drainage technique.