We propose a mass-conservative and monotonicity-preserving characteristic finite element method for solving three-dimensional transport and incompressible Navier-Stokes equations on unstructured grids. The main idea in the proposed algorithm consists of combining a mass-conservative and monotonicity-preserving modified method of characteristics for the time integration with a mixed finite element method for the space discretization. This class of computational solvers benefits from the geometrical flexibility of the finite elements and the strong stability of the modified method of characteristics to accurately solve convection-dominated flows using time steps larger than its Eulerian counterparts. In the current study, we implement three-dimensional limiters to convert the proposed solver to a fully mass-conservative and essentially monotonicity-preserving method in addition of a low computational cost. The key idea lies on using quadratic and linear basis functions of the mesh element where the departure point is localized in the interpolation procedures. The proposed method is applied to well-established problems for transport and incompressible Navier-Stokes equations in three space dimensions. The numerical results illustrate the performance of the proposed solver and support its ability to yield accurate and efficient numerical solutions for three-dimensional convection-dominated flow problems on unstructured tetrahedral meshes.