A model is developed to describe the dynamic forces acting between two deformable drops, or between one drop and a solid surface, when they are in relative axisymmetric motion at separations of less than or equal to 100 nm in a Newtonian liquid. Forces arise from hydrodynamic pressure in the draining liquid film that separates the interfaces and from disjoining pressure due to repulsive or attractive surface forces. Predictions of the model are successfully compared with recent experimental measurements of the force between two micrometer-scale surfactant stabilized decane drops in water in an atomic force microscope (S. L. Carnie, D. Y. C. Chan, C. Lewis, R. Manica, and R. R. Dagastine, Langmuir 21, 2912 (2005); R. R. Dagastine, R. Manica, S. L. Carnie, D. Y. C. Chan, G. W. Stevens, and F. Grieser, Science 313, 210 (2006)) and with subnanometer resolution measurements of time-dependent deformations of a millimeter-scale mercury drop approaching a flat mica surface in a modified surface force apparatus (J. N. Connor and R. G. Horn, Faraday Discuss. 123, 193 (2003); R. G. Horn, M. Asadullah, and J. N. Connor, Langmuir 22, 2610 (2006)). Special limits of the model applicable to small and moderate deformation regimes are also studied to elucidate the key physical ingredients that contribute to the characteristic behavior of dynamic collisions involving fluid interfaces.