Stacking fault tetrahedron (SFT) is a kind of detrimental three-dimensional defect in conventional face-centered cubic (FCC) structural metals; however, its formation and anisotropic mechanical behavior in a CoCrFeNiMn high-entropy alloy (HEA) remain unclear. In this work, we first performed molecular dynamics simulations to verify the applicability of the Silcox-Hirsch mechanism in the CoCrFeNiMn HEA. The mechanical responses of the SFT to shear stress in different directions and that of the pure Ni counterpart were simulated, and the evolutions of the atomic structures of the SFTs during shear were analyzed in detail. Our results revealed that the evolution of the SFT has different patterns, including the annihilation of stacking faults, the formation and expansion of new stacking faults, and insignificant changes in stacking faults. It was found that the effects of SFT on the elastic properties of Ni and HEA are negligible. However, the introduction of SFT would reduce the critical stress, while the critical stress of the CoCrFeNiMn HEA is much less sensitive to SFT than that of Ni.