Honeycomb photonic lattices  share not only many common features with electronic grapheme (a monolayer of carbon atoms arranged in a honeycomb geometry), but can be used to explore phenomena far beyond the original electronic system. Of particular interest are complex gain/loss systems, which, under special conditions, may exhibit complex, but PT-symmetric, Hamiltonians. PT-symmetric systems are characterized by a complex potential, which has neither parity symmetry nor time-reversal symmetry, but is nevertheless symmetric in the product of both . Under these conditions, the eigenvalues of the Hamiltonian are real, in spite of the fact that the potential is complex . Recently, such systems were introduced into the domain of optics . Their simplest realization occurs for two coupled identical waveguides, one with gain and the other with loss, such that the real part of the refractive index is symmetric whereas the imaginary counterpart is anti-symmetric. This realization was recently demonstrated in experiments . Here, we show that adding gain/loss to a regular photonic honeycomb lattice can never result in PT-symmetry. However, this unique system can support the formation of optical tachyons - a photonic version of hypothetical particles with imaginary mass and a group velocity exceeding the vacuum speed of light. Nevertheless, applying a strain to the honeycomb lattice may restore PT-symmetry, in particular in the most interesting region of the band structure: the Dirac regime.