The googleable literature on the Reissner-Nordström repulsion is leaving me confused. This seems to be a contentious topic. I've found the following:
1. Grøn, "Non-existence of the Reissner-Nordström repulsion in general relativity," 1983, "The radial motion of a free neutral test particle in the Reissner-Nordström space-time is studied. It is shown that when the contribution of the electrical field energy to the mass of the central body is properly taken account of, there is no repulsion of the test particle inside the inner horizon of the space-time." http://adsabs.harvard.edu/abs/1983PhLA...94..424G
2. Qadir, "Reissner-Nordstrom repulsion," "It is pointed out that there was an error in the recent paper by Grøn claiming that there is no electro-gravitic repulsion in the Reissner-Nordstrom geometry. It is concluded that the earlier result of Mahajan, Qadir and Valanju still stands." http://adsabs.harvard.edu/abs/1983PhLA...99..419Q
3. Grøn, "Poincaré stress and the Reissner-Nordström repulsion," GRG 20 (1988) 123, " The Reissner-Nordström repulsion is shown to be a consequence of Poincaré stresses in a static, charged object. The strong energy condition implies that the Reissner-Nordström repulsions do not stop a neutral test particle, falling freely from rest at infinity in the fields of a charged body, before it hits the surface of the body. However, if the particle falls from rest at a sufficiently small height above the surface of the body, it will not reach the surface due to the Reissner-Nordström repulsion."
4. "Einstein's general theory of relativity: with modern applications in cosmology," Øyvind Grøn, Sigbjørn Hervik, 2007
[homework problem] "Consider a radially infalling neutral particle in the Reissner-Nordström spacetime with M>|Q|. Show that when the particle comes inside the radius r=Q2/m it will feel a repulsion away from r=0 (i.e. that d2r/dtau2<0 for tau the proper time of the particle). Is this inside or outside the outer horizon r+? Show further that the particle can never reach the singularity at r=0."
Unfortunately 1-3 are not freely accessible online. Are 3 and 4 showing that Grøn has changed his mind (possibly being won over by the arguments in 2)? But the specific statements made in 3-4 do not clearly contradict the ones made in 1.
It's not obvious to me that "repulsion" has a unique definition. The acceleration referred to in 4 is a coordinate acceleration, not a proper acceleration, so it's coordinate-dependent. It seems to me that on a general manifold, "repulsion" clearly can't be well defined. E.g., if a body at the north pole of a sphere causes a body at the south pole to accelerate in a certain direction, is that an attraction, or a repulsion? But I suppose it's possible to define "repulsion" in a more unabmiguous way if your spacetime is asymptotically flat...?