In the present work, we complement the existing dataset of PBE electronic structure properties in the QMOF Database with analogous data computed using three other functionals: HLE17 (a high localexchange meta-GGA), HSE06 (a screened-exchange hybrid GGA), and a functional we refer to here as HSE06 * in which the amount of screened Hartree-Fock (HF) exchange of HSE06 has been changed from 25% at short interelectronic distances to 10%. By analyzing the electronic structure properties calculated at these levels of theory, we uncover severe theoretical limitations associated with the more computationally efficient (meta-)GGA density functionals that prevent them from achieving quantitatively -and sometimes qualitatively -accurate band gap predictions for MOFs and coordination polymers with respect to hybrid functionals. Since it is known that different density functional approximations (DFAs) can alter the underlying charge density, we also investigated trends related to the computed partial atomic charges. In general, we find that the different levels of theory predict similar partial atomic charges; however, as compared to PBE, the meta-GGA and screened hybrids tend to shift electron density away from the metal centers and onto the ligand environments. We conclude by using the new electronic structure data to train multi-task and multi-fidelity convolutional neural network models that can predict PBE, HLE17, HSE06, and HSE06 * band gaps given a graph-based representation of a MOF crystal structure. We anticipate that the computational data, trends, and subsequent deep learning models presented in this work will make it possible to achieve both rapid and accurate predictions of MOF band gaps that can greatly accelerate the materials design and discovery process. To help realize this vision, all the data underlying the QMOF Database is now also made available as a dedicated, interactive application on the widely used Materials Project.