To specifically answer the question: The main shortcoming of Density Functional Theory is that even though it is a formally exact reformulation of quantum theory, in the current state of the theory, approximations are required for the Exchange-Correlation energy functional. All the Density-Functional approximations that we have so far fail to exactly reproduce the contributions from different phenomena to the Exchange- and Correlation-Energies.
As discussed by Cohen, Mori-Sánchez, and Yang in a paper titled "Insights into Current Limitations of Density Functional Theory" in the journal Science 2008 most of the weaknesses can be traced back to two main errors of standard density-functionals: The delocalization error and the static correlation error.
One has to read the paper to understand the details, but in a hand-wavy explanation what they say is that when using DFT, the electron density (or electron cloud) is artificially spread-out due to an incorrect behavior of the standard functionals.
This problem has its root in the fact that when using DFT even if you have only one electron, the density of that electron (a non-local object) interacts with the electron itself (a local object) producing an artificial repulsion of the electron caused by itself. An analogous situation happens with spin-spin interaction.
This is an artifact on the formulation of the exchange-energy functional (exact) that the correlation-energy functional (approximated) cannot correct in any of the functionals that we have prepared so far, including the fanciest ones, i.e. the "Minnesota" family.
This is reflected in the underestimation of the barriers of chemical reactions, the band gaps of materials, the energies of dissociating molecular ions, and charge transfer excitation energies. Density-Functional approximations also overestimate the binding energies of charge transfer complexes and the response to an electric field in molecules and materials.
Another practical issue in DFT is that it is not variational, which is a fancy terminology to say that if you use one of the simplest functionals and you get a some answer, you are not guaranteed to improve it by using a more complicated functional. Choosing a functional is a matter of experience and sometimes, luck.
Even though all of this may sound really bad, it surprising how DFT works much better and/or faster than other computational quantum methods modelling many different properties important for physics, chemistry and materials sciences.
For more details, I would also recommend the book by Parr and Yang, Density Functional Theory of Atoms and Molecules.