I don't see why your statement would be true in general for all Green's functions, in fact I have a hunch this not the case for two-particle Green's functions such as current-current correlation functions, but let's simply work with one-particle Green's functions to show why this is true.
Consider a many-electron system. It could be an insulator with the chemical potential in the band gap, or it could be a conductor. Now consider the following one-particle Green's functions, where $c_k$ creates an electron with momentum $k$ in a state above the Fermi energy:
$$G^<(k,t)=i\langle c_k^\dagger(0) c_k(t)\rangle$$
At zero temperature, the conduction band state created $c_k^\dagger$ is empty, such that
Therefore, at zero temperature
One can also show that the definition of $G^R(k,t)$ given here is equivalent to your definition above. So we have determined that the retarded and time-ordered Green's functions are equal for one-particle excitations above the Fermi level, i.e. $\omega>0$ excitations, as long as we're working at zero temperature where the states above the Fermi energy are empty at equilibrium. One could probably be even more formal in proving that the creation of an electron necessarily raises the energy of the system, but this seems to be intuitively plausible.
Now why is this result only true for $\omega>0$? This is because if we instead create a hole in a state below the Fermi energy, we find that instead
such that $G^R(k,t)$ is still nonzero only at positive times, while $G^T(k,t)$ is now only nonzero at negative times, and these functions are now clearly different. The analog of your claim for negative energy excitations is that the advanced Green's function and time-ordered Green's function are identical for negative energy excitations.
Now you may be using a different definition of operators in your Green's functions, i.e. $$c_k \rightarrow \psi(r)$$
however you should find that the statements made here generalize when you write spatial Green's functions in terms of the momentum Green's functions discussed here.
In this way you'll find that a spatial Green's function can be written in terms of the electron and hole Green's functions described here. The poles of the time-ordered Green's function will match those of the retarded Green's function at positive energies and those of the advanced Green's function at negative energies.