The Lattice Boltzmann Method (LBM) has established itself as a popular numerical method in computational fluid dynamics. Several advancements have been recently made in LBM, which include multiple-relaxation-time LBM to simulate anisotropic advection-diffusion processes. Because of the importance of LBM simulations for transport problems in subsurface and reactive flows, one needs to study the accuracy and structure preserving properties of numerical solutions under the LBM. The solutions to advective-diffusive systems are known to satisfy maximum principles, comparison principles, the non-negative constraint, and the decay property. In this paper, using several numerical experiments, it will be shown that current single- and multiple-relaxation-time lattice Boltzmann methods fail to preserve these mathematical properties for transient diffusion-type equations. We will also show that these violations may not be removed by simply refining the discretization parameters. More importantly, it will be shown that meeting stability conditions alone does not guarantee the preservation of the aforementioned mathematical principles and physical constraints in the discrete setting. A discussion on the source of these violations and possible approaches to avoid them is included. A condition to guarantee the non-negativity of concentration under LBM in the case of isotropic diffusion is also derived. The impact of this research is twofold. First, the study poses several outstanding research problems, which should guide researchers to develop LBM-based formulations for transport problems that respect important mathematical properties and physical constraints in the discrete setting. This paper can also serve as a good source of benchmark problems for such future research endeavors. Second, this study cautions the practitioners of the LBM for transport problems with the associated numerical deficiencies of the LBM, and provides guidelines for performing predictive simulations of advective-diffusive processes using the LBM.