that the Casimir force (which arises from quantum electrodynamic fluctuations [@casimir; @Lifshitz80; @milton04]) can enable dielectric objects (micro-spheres) with certain material dispersions to form stable non-touching configurations (diclusters) in fluids [@RodriguezMc10:PRL; @Rodriguez08:PRL]. Such micro-sphere interactions are predicted to possess a variety of unusual Casimir effects, including repulsive forces, [@Kenneth; @Munday09; @Dzyaloshinskii61] a strong interplay with material dispersion [@RodriguezMc10:PRL], and strong temperature dependences [@RodriguezW10:PRL], and may have applications in microfluidic particle suspensions [@RahiZa09:arxiv; @McCauleyRo10:PRA]. A typical situation considered in this paper is depicted in [Fig. \[fig:fig1\]]{}, consisting of silicon and teflon micro-spheres suspended in ethanol above a gold substrate. Although our earlier work considered pairs of micro-spheres suspended above a substrate in the additive/pairwise approximation, summing the exact two–body sphere–sphere and sphere–substrate interactions, in this paper we perform exact three-body calculations. In [Sec. \[sec:pairwise\]]{}, we explicitly demonstrate the breakdown of the pairwise approximation for sufficiently small spheres, in which an adjacent substrate modifies the equilibrium sphere separation, but we also identify experimentally relevant regimes in which pairwise approximations (and even a parallel-plate proximity-force/PFA approximation [@Derjaguin]) are valid. In [Sec. \[sec:temp\]]{}, we also consider temperature corrections to the Casimir interactions. Although a careful choice of materials can lead to a large temperature 