A standard plane-wave expansion technique is used to investigate the evolution of the photonic band structure of a two-dimensional honeycomb lattice composed by cylindrical shell rods with dielectric permittivities ε1 and ε2, and embedded in a background with permittivity ε3. We have considered the effect of dispersive dielectric responses as well as the influence of an externally applied magnetic field aiming to obtain efficient tunable bandgaps. Present results suggest that a combination of a doped semiconductor constituent with an anisotropic geometry, which breaks symmetry and unfolds degeneracies, provides an efficient realization of photonic systems with tunable bandgaps.