Using a standard technique of Li and Yau, we study heat kernel estimates for a special type of compact conformally Kähler manifold, called a multiplier Hermitian manifold of type $\sigma$, which we derive from a Hamiltonian holomorphic vector field on the manifold. In particular, we obtain a lower bound estimate for the Green function averaged by the associated group action. For a fixed $\sigma$, such an estimate is known to play a crucial role in the proof of the uniqueness, modulo a group action, of Einstein multiplier Hermitian structures on a given Fano manifold.