In this paper, we introduce a nonlinear finite volume scheme preserving discrete strong extremum principle (DSEP) for diffusion equations on tetrahedral meshes. In the construction of our nonlinear scheme, the key is to reformulate a discrete normal flux with local extremum principle structure, which is based on a modification of a second order linear scheme. In the construction of existing cell-centered finite volume schemes that maintain the discrete maximum principle, it is required to represent auxiliary unknowns as convex combinations of primary unknowns, which results in strong constraints on the smoothness of the mesh and diffusion coefficient. By contrast, our new scheme avoids this kind of constraints. Moreover, we will prove that there holds the DSEP for any solution of our scheme and there exists at least one solution preserving DSEP for our scheme. Furthermore, a modified Picard iteration with the Anderson acceleration (mP-AA) for solving the nonlinear scheme is proposed, and the nonlinear convergence of the modified Picard iteration is also proved. Finally, numerical examples are presented to show that the new scheme preserves DSEP and obtains second order accuracy, as well as the mP-AA method is effective.