  Motivated by recent experiments (Subero et. al. Nature Comm. $\bf{14}$, 7924
(2023)) we analyze photonic heat transport through a Josephson junction in a
dissipative environment. For this purpose we derive the general expressions for
the heat current in terms of non-equilibrium Green functions for the junction
coupled in series or in parallel with two environmental impedances at different
temperatures. We show that even on the insulating side of the Schmid transition
the heat current is sensitive to the Josephson coupling exhibiting an opposite
behavior for the series and parallel connection and in qualitative agreement
with experiments. We also predict that this device should exhibit heat
rectification properties and provide simple expressions to account for them in
terms of the system parameters.
