The Casimir-Polder force between two atoms with equal uniform acceleration and separated by a constant distance R is considered. We show that, in the low-acceleration limit, while the near-zone R-6 behavior of the interatomic interaction energy is not changed by the acceleration of the atoms, the far-zone interaction energy decreases as R-5 instead of the well-known R-7 behavior for inertial atoms. Possibility of an indirect detection of the Unruh effect through measurements of the Casimir-Polder force between the two accelerating atoms is also suggested. We also consider a heuristic model for calculating the Casimir-Polder potential energy between the two atoms in the high-acceleration limit.