In the framework of classical field theory we consider some new topological effects in (2 + 1)-dimensional Einstein gravity. The investigation is based on the explicit expression for the Euclidean Green function for massless fields (scalar with minimal coupling or electrostatic) on the two-dimensional Riemannian surface. Expressions for the topological self-energy of a point charge and point dipole are obtained. These results are interpreted in electrostatics terms on the Euclidean plane. The applications to the cases of (2 + 1)-dimensional conical and multiconical spaces and to the spacetime of a circular dust string and spherical star in (2 + 1) dimensions are considered. We have found the existence of an additional interaction between charged particles which depends on the topology of the Riemannian surface. The same effect arises when we consider two uncharged gravitating particles in the presence of classical charged matter. Both these effects may be considered as a Casimir-like interaction in classical theory.