Dissipation of mechanical energy underlies the sensitivity of many nanomechanical devices, with environmental effects often having a significant effect. One case of practical relevance is the interaction of elastic beam resonators with fluid, which is known to dramatically increase energy dissipation. Recently, we investigated energy dissipation in a different class of elastic beam resonator that embeds a microfluidic channel in its interior. In this paper, we examine the effect of the beam material Poisson ratio on these devices and discover that it can strongly affect energy dissipation—this is in direct contrast to conventional cantilever beams immersed in fluid. Increasing the Poisson ratio in these microfluidic devices is found to decrease energy dissipation, with the incompressible material limit providing minimum energy dissipation. Our paper establishes that, in this limit, placement of the fluid channel away from the beam neutral axis has negligible effect on energy dissipation in many cases of practical interest. The physical implications of these findings are discussed, and a detailed comparison with available experimental results is provided.