= Solution
Changing the target metric by $\delta G_{\mu\nu}=\epsilon_{\mu\nu}$ changes the action by
$$
\delta S=-\frac1{4\pi\alpha'}\int_\Sigma d^2\sigma\sqrt{|h|}\,
h^{ab}\epsilon_{\mu\nu}\partial_aX^\mu\partial_bX^\nu.
$$
Since the path-integral weight is $e^{iS}$, the leading change is the insertion
$$
\delta\langle\phi_1\cdots\phi_n\rangle
=\int_\Sigma d^2\sigma\,
\langle\mathcal O(\sigma,\tau)\phi_1\cdots\phi_n\rangle_{\rm conn},
$$
where
$$
\boxed{\mathcal O=-\frac{i}{4\pi\alpha'}\sqrt{|h|}\,
h^{ab}\epsilon_{\mu\nu}\partial_aX^\mu\partial_bX^\nu}.
$$
A change of worldsheet metric within a gauge orbit changes only the redundant description and cannot alter gauge-invariant observables. The target-space metric is instead a physical coupling of the <string nonlinear sigma model>, so changing it changes the theory and its correlation functions.
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