Motivated by recent developments in twisted bilayer graphene moire superlattices, we investigate the effects of electron-electron interactions in a honeycomb lattice with an applied periodic potential using a finite-temperature Wilson-Fisher momentum shell renormalization group (RG) approach. We start with a low-energy effective theory for such a system, at first giving a discussion of the most general case in which no point group symmetry is preserved by the applied potential, and then focusing on the special case in which the potential preserves a D-3 point group symmetry. As in similar studies of bilayer graphene, we find that, while the coupling constants describing the interactions diverge at or below a certain critical temperature T = T-c, it turns out that ratios of these constants remain finite and in fact provide information about what types of orders the system is becoming unstable to. However, in contrast to these previous studies, we only find isolated fixed rays, indicating that these orders are likely unstable to perturbations to the coupling constants. Our RG analysis leads to the qualitative conclusion that the emergent interaction-induced symmetry-breaking phases in this model system, and perhaps therefore by extension in twisted bilayer graphene, are generically unstable and fragile, and may thus manifest strong sample dependence.