Recently synthesized two-dimensional covalent organic frameworks (COFs) exhibit high surface area, large pore size, and unique structural architectures, making them promising materials for various energy applications. Here, a total of nine COFs structures, including two deposited on a hexagonal boron nitride substrate, are investigated using density functional theory, quasi-particle many-body theory within the GW approximation, and an image charge model. The structures considered belong to two major families (thiophene-based COF-n (T-COF-n) and tetrakis (4-aminophenyl) porphyrin-x (TAPP-x)) differing from the presence of B-O or C=N linkers. While T-COF-n structures are shown to constitute planar networks, TAPP-x systems can display non-negligible corrugation due to the out-of-plane rotation of phenyl rings. We find that the electronic properties do not differ significantly when altering the chain molecules within each family. Many-body effects are shown to lead to large band-gap increase while the presence of the substrate yields appreciable reductions of the gaps, due to substrate polarization effects.