I once worked this out studying Polchinski's book, to which I refer for notation and conventions. I reproduce my work here verbatim.
We do this in separate steps, starting with the variation of the dilaton field
\begin{align}
\delta_\Phi {S} =&\, \frac{1}{2\kappa_0^2} \int d^Dx\, \delta_\Phi \Bigg\{ \sqrt{-G}\, e^{-2\Phi} \Big[ -\frac{2(D-26)}{3\alpha'} + {R} - \frac{1}{12} H_{\mu\nu\lambda}H^{\mu\nu\lambda} + 4 \partial_\mu \Phi \partial^\mu \Phi \Big] \Bigg\}\nonumber\\
=&\, \frac{1}{2\kappa_0^2} \int d^Dx\, \Bigg\{ e^{-2\Phi} (-2\delta\Phi) \sqrt{-G}\, \Big[ -\frac{2(D-26)}{3\alpha'} + {R} - \frac{1}{12} H_{\mu\nu\lambda}H^{\mu\nu\lambda} + 4 \partial_\mu \Phi \partial^\mu \Phi \Big] \nonumber\\
&\qquad \qquad+ \sqrt{-G}\, e^{-2\Phi} 8 \partial_\mu \Phi \partial^\mu \delta \Phi \Bigg\}\nonumber\\
=&\, \frac{1}{2\kappa_0^2} \int d^Dx\, \Bigg\{ -2e^{-2\Phi} \sqrt{-G}\, \Big[ -\frac{2(D-26)}{3\alpha'} + {R} - \frac{1}{12} H_{\mu\nu\lambda}H^{\mu\nu\lambda} + 4 \partial_\mu \Phi \partial^\mu \Phi \Big] \nonumber\\
&\qquad \qquad-8 \partial^\mu \Big[ \sqrt{-G}\, e^{-2\Phi} \partial_\mu \Phi\Big]\Bigg\} \delta \Phi \nonumber
\end{align}
\begin{align}
\phantom{\delta_\Phi {S}}
=&\, -\frac{1}{\kappa_0^2} \int d^Dx\, e^{-2\Phi} \sqrt{-G}\, \Bigg[ -\frac{2(D-26)}{3\alpha'} + {R} - \frac{1}{12} H_{\mu\nu\lambda}H^{\mu\nu\lambda} + 4 \partial_\mu \Phi \partial^\mu \Phi \nonumber\\
&\qquad \qquad+ 4(-G)^{-1/2}\big( \partial^\mu\sqrt{-G}\big) \partial_\mu \Phi +4 \big(-2\partial^\mu \Phi\big) \partial_\mu\Phi +4 \partial^\mu \partial_\mu \Phi \Bigg] \delta \Phi \nonumber\\
=&\, -\frac{1}{\kappa_0^2} \int d^Dx\, e^{-2\Phi} \sqrt{-G}\, \Bigg[ -\frac{2(D-26)}{3\alpha'} + {R} - \frac{1}{12} H_{\mu\nu\lambda}H^{\mu\nu\lambda} -4 \partial_\mu \Phi \partial^\mu \Phi \nonumber\\
&\qquad \qquad+2 G^{\nu\sigma}\partial^\mu G_{\nu\sigma} \partial_\mu \Phi +4 \partial_\mu \partial^\mu \Phi \Bigg] \delta \Phi \nonumber\\
=&\, -\frac{1}{2\kappa_0^2 \alpha'} \int d^Dx\, e^{-2\Phi} \sqrt{-G}\, 2\delta\Phi \Bigg\{ -4\Bigg[ \frac{D-26}{6} +\alpha' \partial_\mu \Phi \partial^\mu \Phi -\frac{\alpha'}{2}\partial_\mu \partial^\mu \Phi -\frac{\alpha'}{24} H_{\mu\nu\lambda}H^{\mu\nu\lambda} \Bigg]\nonumber\\
&\qquad \qquad+ \alpha'{R} + 2 \alpha' \partial^\mu\partial_\mu \Phi-\frac{\alpha'}{4} H_{\mu\nu\lambda}H^{\mu\nu\lambda} \Bigg\} \nonumber\\
=&\, -\frac{1}{2\kappa_0^2 \alpha'} \int d^Dx\, e^{-2\Phi} \sqrt{-G}\, 2\delta\Phi \Big( -4 \beta^\Phi + \beta^{G\, \mu}_{\;\;\mu} \Big)
\end{align}
In the last line we have replaced the ordinary derivative by the covariant derivative, used the fact that the spacetime metric is covariantly constant, $\nabla_\mu G^{\mu\nu}=0$ and used the definition in (3.7.14).
Let us now consider
\begin{align}
\delta_B {S} =&\, \frac{1}{2\kappa_0^2} \int d^Dx\, \delta_B \Bigg\{ \sqrt{-G}\, e^{-2\Phi} \Big[ -\frac{2(D-26)}{3\alpha'} + {R} - \frac{1}{12} H_{\mu\nu\lambda}H^{\mu\nu\lambda} + 4 \partial_\mu \Phi \partial^\mu \Phi \Big] \Bigg\}\nonumber\\
=&\, \frac{1}{2\kappa_0^2} \int d^Dx\, \sqrt{-G}\, e^{-2\Phi} \left( -\frac{1}{6} H^{\mu\nu\lambda} \delta_B H_{\mu\nu\lambda} \right)=-\frac{1}{4\kappa_0^2} \int d^Dx\, \sqrt{-G}\, e^{-2\Phi} H^{\mu\nu\lambda} \delta_B \partial_\mu B_{\nu\lambda} \nonumber\\
=&\, \frac{1}{4\kappa_0^2} \int d^Dx\, \partial_\mu \left(\sqrt{-G}\, e^{-2\Phi} H^{\mu\nu\lambda}\right) \delta B_{\nu\lambda} \nonumber\\
=&\, \frac{1}{4\kappa_0^2} \int d^Dx\, \sqrt{-G}\, e^{-2\Phi} \Big(+\frac{1}{2} G^{\nu\sigma} \partial_\mu G_{\nu\sigma} H^{\mu\nu\lambda} -2 \partial_\mu \Phi H^{\mu\nu\lambda} + \partial_\mu H^{\mu\nu\lambda} \Big) \delta B_{\nu\lambda} \nonumber\\
=&\, -\frac{1}{2\kappa_0^2 \alpha'} \int d^Dx\, \sqrt{-G}\, e^{-2\Phi} \Big(-\frac{\alpha'}{4} G^{\nu\sigma} \partial_\mu G_{\nu\sigma} H^{\mu\nu\lambda} +\alpha' \partial_\mu \Phi H^{\mu\nu\lambda} -\frac{\alpha'}{2} \partial_\mu H^{\mu\nu\lambda} \Big) \delta B_{\nu\lambda} \nonumber\\
=&\, -\frac{1}{2\kappa_0^2 \alpha'} \int d^Dx\, \sqrt{-G}\, e^{-2\Phi} \beta^B_{\mu\nu} \delta B_{\nu\lambda}
\end{align}
Here also, we have replaced the ordinary derivative by the covariant derivative, used the fact that the spacetime metric is covariantly constant, $\nabla_\mu G^{\mu\nu}=0$ and used the definition in (3.7.14).
Finally we consider
\begin{align}
\delta_G {S} =&\, \frac{1}{2\kappa_0^2} \int d^Dx\, \delta_G \Bigg\{ \sqrt{-G}\, e^{-2\Phi} \Big[ -\frac{2(D-26)}{3\alpha'} + {R} - \frac{1}{12} H_{\mu\nu\lambda}H^{\mu\nu\lambda} + 4 \partial_\mu \Phi \partial^\mu \Phi \Big] \Bigg\}\end{align}
Recall that the variation of the Einstein-Hilbert action is
\begin{align}
\delta_G \int d^Dx\, \sqrt{-G}\, {R} = \int d^D x \, \sqrt{-G}\left( {R}_{\mu\nu}-\frac{1}{2} G_{\mu\nu}{R}\right) \delta G^{\mu\nu}
\end{align}
But we need to be careful as we have an extra factor $e^{-2\Phi}$. We split the calculation
\begin{align}
\delta_G \int d^Dx\, \sqrt{-G}\, e^{-2\Phi} G^{\mu\nu} {R}_{\mu\nu} =& \sum_{a=1}^4 \delta_G I_a
\end{align}
with
\begin{align}
\delta_G I_1 =&\, \int d^Dx\, (\delta_G \sqrt{-G}) \, e^{-2\Phi} G^{\mu\nu} {R}_{\mu\nu} = \int d^Dx \, \frac{1}{2} \sqrt{-G} G^{\mu\nu} \delta G_{\mu\nu} e^{-2\Phi} G^{\rho\sigma} {R}_{\rho\sigma} \nonumber\\
=&\, \int d^Dx \sqrt{-G} e^{-2\Phi} \Big[ \frac{1}{2} G^{\mu\nu} {R} \Big]\delta G_{\mu\nu}
\end{align}
Next,
\begin{align}
\delta_G I_2 =&\, \int d^Dx\, \sqrt{-G} \, (\delta_G e^{-2\Phi}) G^{\mu\nu} {R}_{\mu\nu} = 0
\end{align}
Third,
\begin{align}
\delta_G I_3 =&\, \int d^Dx\, \sqrt{-G} \, e^{-2\Phi}(\delta_G G^{\mu\nu} ) {R}_{\mu\nu} = \int d^Dx\, \sqrt{-G} \, e^{-2\Phi}(- G^{\mu\rho} G^{\nu\sigma} \delta G_{\rho\sigma } ){R}_{\mu\nu} \nonumber\\
=&\, \int d^Dx \sqrt{-G} e^{-2\Phi} \Big[-{R}^{\mu\nu} \Big]\delta G_{\mu\nu}
\end{align}
Finally, and this is where the change occurs compared to the Einstein-Hilbert action
\begin{align}
\delta_G I_4 =&\, \int d^Dx\, \sqrt{-G} \, e^{-2\Phi } G^{\mu\nu}\delta_G {R}_{\mu\nu} =
\end{align}
The variation of the Ricci tensor is
$\delta {R}_{\mu \nu} = \delta {R}^\rho_{\mu \rho\nu } = \nabla_\rho \delta \Gamma^\rho_{\mu \nu} - \nabla_\nu \delta \Gamma^\rho_{\mu \rho} $. Thus
\begin{align}
\delta_G I_4 =&\, \int d^Dx\, \sqrt{-G} \, e^{-2\Phi} G^{\mu\nu }( \nabla_\rho \delta_G\Gamma^\rho_{\mu \nu} - \nabla_\nu \delta_G \Gamma^\rho_{\mu \rho} )
\nonumber\\
=& \, \int d^Dx\, e^{-2\Phi} \big[ \sqrt{-G} \, \nabla_\rho(G^{\mu\nu } \delta_G\Gamma^\rho_{\mu \nu}) - \sqrt{-G} \, \nabla_\nu (G^{\mu\nu } \delta_G \Gamma^\rho_{\mu \rho}) \big] \nonumber\\
=& \, \int d^Dx\, e^{-2\Phi} \big[ \partial_\rho(\sqrt{-G} \, G^{\mu\nu } \delta_G\Gamma^\rho_{\mu \nu}) - \partial_\nu (\sqrt{-G} \, G^{\mu\nu } \delta_G \Gamma^\rho_{\mu \rho}) \big]
\end{align}
In the case of the Einstein-Hilbert action, i.e. $\Phi=0$, this is a total derivative and vanishes. This times this is not the case as we get a contribution from the dilaton field upon partial integration
\begin{align}
\delta_G I_4=&\, 2 \int d^Dx\, e^{-2\Phi} \sqrt{-G} \, G^{\mu\nu } \big[ \partial_\rho \Phi\delta_G\Gamma^\rho_{\mu \nu} - \partial_\nu \Phi \delta_G \Gamma^\rho_{\mu \rho} \big]
\end{align}
Unfortunately, this time we need to work out the variations of the connections
\begin{align}
\delta_G \Gamma^\rho_{\mu\nu} =&\, \delta_G \frac{1}{2} G^{\rho\sigma}(\partial_\mu G_{\sigma\nu} + \partial_\nu G_{\sigma\mu} - \partial_\sigma G_{\mu\nu}) \nonumber\\
=&\, \frac{1}{2}\Big[ - G^{\rho\kappa} G^{\sigma\tau} \delta G_{\kappa\tau} (\partial_\mu G_{\sigma\nu} + \partial_\nu G_{\sigma\mu} - \partial_\sigma G_{\mu\nu}) \nonumber\\
&+
G^{\rho\sigma}(\partial_\mu \delta G_{\sigma\nu} + \partial_\nu \delta G_{\sigma\mu} - \partial_\sigma \delta G_{\mu\nu}) \Big]
\end{align}
Recall that by covariance we can, to that order, replace all partial derivatives by covariant derivatives. This means that we can ignore the first line. We then split the calculation in two
\begin{align}
\delta_G I_{4a} = &\, 2 \int d^Dx\, e^{-2\Phi} \sqrt{-G} \, G^{\mu\nu } \partial_\rho \Phi \frac{1}{2} G^{\rho\sigma}(\partial_\mu \delta G_{\sigma\nu} + \partial_\nu \delta G_{\sigma\mu} - \partial_\sigma \delta G_{\mu\nu}) \nonumber\\
=&\, - \int d^Dx\, \sqrt{-G} \, G^{\mu\nu} G^{\rho\sigma} \Big[ \partial_\mu \left( e^{-2\Phi} \partial_\rho\Phi \right) \delta G_{\sigma\nu} +
\partial_\nu \left( e^{-2\Phi} \partial_\rho\Phi \right) \delta G_{\sigma\mu}\nonumber\\
& - \partial_\sigma \left( e^{-2\Phi} \partial_\rho \Phi\right) \delta G_{\mu\nu}\Big]\nonumber\\
=&\, - \int d^Dx\, \sqrt{-G} \, G^{\mu\nu} G^{\rho\sigma} \Big[2 \partial_\mu \left( e^{-2\Phi} \partial_\rho \Phi\right) \delta G_{\sigma\nu} - \partial_\sigma \left( e^{-2\Phi} \partial_\rho\Phi \right) \delta G_{\mu\nu}\Big]\nonumber\\
=&\, - \int d^Dx\, \sqrt{-G} \, \partial_\sigma \left( e^{-2\Phi} \partial_\rho\Phi\right) (2G^{\sigma\nu} G^{\rho\mu} - G^{\mu\nu} G^{\rho\sigma}) \delta G_{\mu\nu}
\end{align}
The second part is
\begin{align}
\delta_G I_{4b} = -&\, 2 \int d^Dx\, e^{-2\Phi} \sqrt{-G} \, G^{\mu\nu } \partial_\nu \Phi \frac{1}{2} G^{\rho\sigma}(\partial_\mu \delta G_{\sigma\rho} + \partial_\rho \delta G_{\sigma\mu} - \partial_\sigma \delta G_{\rho\mu}) \nonumber\\
=&\, \int d^Dx\, \sqrt{-G} \, G^{\mu\nu } G^{\rho\sigma} \Big[ \partial_\mu \left( e^{-2\Phi} \partial_\nu\Phi \right) \delta G_{\rho \sigma} + \partial_\rho \left( e^{-2\Phi} \partial_\nu\Phi \right) \delta G_{ \sigma \mu} \nonumber\\
&- \partial_\sigma \left( e^{-2\Phi} \partial_\nu\Phi \right) \delta G_{ \rho \mu} \Big]\nonumber\\
=&\, \int d^Dx\, \sqrt{-G} \, G^{\mu\nu } G^{\rho\sigma} \partial_\mu \left( e^{-2\Phi} \partial_\nu\Phi \right) \delta G_{\rho \sigma} \nonumber\\
=&\, \int d^Dx\, \sqrt{-G} \, \partial_\sigma \left( e^{-2\Phi} \partial_\rho \Phi \right)G^{\mu\nu } G^{\rho\sigma} \delta G_{\mu\nu}
\end{align}
Therefore
\begin{align}
\delta_G I_{4}
=&\, \int d^Dx\, \sqrt{-G} \, \partial_\sigma \left( e^{-2\Phi} \partial_\rho \Phi \right)(- 2G^{\sigma\nu} G^{\rho\mu} + G^{\mu\nu} G^{\rho\sigma} + G^{\mu\nu } G^{\rho\sigma})\delta G_{\mu\nu} \nonumber\\
=&\, \int d^Dx\, \sqrt{-G} \, \partial_\sigma \left( e^{-2\Phi} \partial_\rho \Phi \right)2(G^{\mu\nu} G^{\rho\sigma} - G^{\sigma\nu} G^{\rho\mu} )\delta G_{\mu\nu} \nonumber\\
=&\, \int d^Dx\, \sqrt{-G} \, e^{-2\Phi} (-2 \partial_\sigma \Phi\partial_\rho \Phi +\partial_\rho\partial_\sigma \Phi )\, 2(G^{\mu\nu} G^{\rho\sigma} - G^{\sigma\nu} G^{\rho\mu} )\delta G_{\mu\nu} \nonumber\\
=&\, \int d^Dx\, \sqrt{-G} \, e^{-2\Phi} (-4 \partial_\sigma \Phi\partial^\sigma \Phi G^{\mu\nu} + 4 \partial^\mu \Phi\partial^\nu \Phi +2\partial_\sigma \partial^\sigma \Phi G^{\mu\nu }- 2\partial^\mu \partial^\nu \Phi) \delta G_{\mu\nu}
\end{align}
Adding the four pieces together we find
\begin{align}
\delta_G \int d^Dx\, \sqrt{-G}\, e^{-2\Phi} G^{\mu\nu} {R}_{\mu\nu} =&\, \int d^Dx \sqrt{-G} e^{-2\Phi} \Big( \frac{1}{2} G^{\mu\nu} {R} -{R}^{\mu\nu} \nonumber\\
& -4 \partial_\sigma \Phi\partial^\sigma \Phi G^{\mu\nu} + 4 \partial^\mu \Phi\partial^\nu \Phi +2\partial_\sigma \partial^\sigma \Phi G^{\mu\nu }- 2\partial^\mu \partial^\nu \Phi
\Big)\delta G_{\mu\nu}
\end{align}
As expected this gives back the Einstein equations when we have a constant $\Phi$, but we see that the dilaton field gives a correction that includes its derivative only.
There is another thing that we need to be careful about as well. Any indices upstairs have been raised via the spacetime ``metric'' $G^{\mu\nu}$ so they also carry a metric dependence.
For example
\begin{align}
\delta_G H_{\mu\nu\lambda}H^{\mu\nu\lambda} =&\, H_{\mu\nu\lambda}\delta_G ( G^{\mu\sigma} G^{\nu\rho} G^{\lambda\kappa} H_{\sigma\rho\kappa}) = 3 H_{\mu\nu\lambda} G^{\mu\sigma} G^{\nu\rho} H_{\sigma\rho\kappa} \delta G^{\lambda\kappa} \nonumber\\
=&\, -3 H_{\mu\nu\lambda} G^{\mu\sigma} G^{\nu\rho} H_{\sigma\rho\kappa}G^{\lambda\tau} G^{\kappa\eta} \delta G_{\tau\eta}\nonumber\\
=&\, -3 H^{\sigma\rho\tau} H^\eta_{\;\;\rho\sigma}\delta G_{\tau\eta}= -3 H^{\mu\rho\sigma} H^\nu_{\;\;\rho\sigma}\delta G_{\mu\nu}
\end{align}
Similarly
\begin{align}
\delta_G \partial_\mu\Phi \partial^\mu \Phi = &\, \delta_G G^{\mu\nu} \partial_\mu\Phi \partial_\nu \Phi = - G^{\mu\rho} G^{\nu\sigma} \partial_\mu\Phi \partial_\nu \Phi \delta G_{\rho\sigma} = - \partial^\mu \Phi \partial^\nu \Phi \delta G_{\mu\nu}
\end{align}
So we have
\begin{align}
\delta_G {S} =&\, \frac{1}{2\kappa_0^2} \int d^Dx\, e^{-2\Phi} \Bigg\{ \delta_G \big( \sqrt{-G}\big) \Big[ -\frac{2(D-26)}{3\alpha'} - \frac{1}{12} H_{\rho\sigma\lambda}H^{\rho\sigma\lambda} + 4 \partial_\sigma \Phi \partial^\sigma \Phi \Big]\nonumber\\
& + \sqrt{-G}\Big( -{R}^{\mu\nu}+\frac{1}{2} G^{\mu\nu}{R} -4 \partial_\sigma \Phi\partial^\sigma \Phi G^{\mu\nu} + 4 \partial^\mu \Phi\partial^\nu \Phi +2\partial_\sigma \partial^\sigma \Phi G^{\mu\nu }- 2\partial^\mu \partial^\nu \Phi \nonumber\\
& \qquad \qquad+\frac{1}{4} H^{\mu\rho\sigma} H^\nu_{\;\;\rho\sigma} -4 \partial^\mu \Phi \partial^\nu \Phi \Big)\delta G_{\mu\nu} \Bigg\} \nonumber\\
=&\, \frac{1}{2\kappa_0^2} \int d^Dx\, \sqrt{-G}\, e^{-2\Phi} \Bigg\{ \frac{1}{2} G^{\mu\nu} \delta G_{\mu\nu} \Big[ -\frac{2(D-26)}{3\alpha'} - \frac{1}{12} H_{\rho\sigma\lambda}H^{\rho\sigma\lambda} + 4 \partial_\sigma \Phi \partial^\sigma \Phi \Big]\nonumber\\
& + \Big( -{R}^{\mu\nu}+\frac{1}{2} G^{\mu\nu}{R} -4 \partial_\sigma \Phi\partial^\sigma \Phi G^{\mu\nu} +2\partial_\sigma \partial^\sigma \Phi G^{\mu\nu }- 2\partial^\mu \partial^\nu \Phi +\frac{1}{4} H^{\mu\rho\sigma} H^\nu_{\;\;\rho\sigma} \Big)\delta G_{\mu\nu} \Bigg\} \nonumber\\
&=\, -\frac{1}{2\kappa_0^2\alpha'} \int d^Dx\, \sqrt{-G}\, e^{-2\Phi} \Big[
\alpha'{R}^{\mu\nu} +2 \alpha' \partial^\mu \partial^\nu \Phi-\frac{\alpha'}{4} H^{\mu\rho\sigma} H^\nu_{\;\;\rho\sigma} \nonumber\\
& -\frac{1}{2} G^{\mu\nu}\Big( \alpha' G^{\mu\nu}{R} +4\alpha'\partial_\sigma \partial^\sigma \Phi-\frac{\alpha'}{12} H_{\rho\sigma\lambda}H^{\rho\sigma\lambda}-\frac{2(D-26)}{3} -4 \alpha' \partial_\sigma \Phi \partial^\sigma \Phi\Big) \Big]
\end{align}
We can rewrite the term between brackets as
\begin{align}
\mathfrak{B}^{\mu\nu} =\beta^{G\; \mu\nu} -\frac{1}{2} G^{\mu\nu} (\beta^{G\; \sigma}_{\;\;\;\sigma} -4\beta^\Phi)
\end{align}
Indeed
\begin{align}
\mathfrak{B}^{\mu\nu} = &\, \alpha' {R}^{\mu\nu} + 2\alpha' \nabla^\mu \nabla^\nu \Phi - \frac{\alpha'}{4} H^{\mu\rho\sigma}H^\nu_{\;\;\rho\sigma} -\frac{1}{2} G^{\mu\nu}\Big(\alpha' {R} + 2\alpha' \nabla^2\Phi - \frac{\alpha'}{4} H^{\lambda\rho\sigma}H_{\lambda\rho\sigma} \nonumber\\
&
- \frac{2(D-26)}{3} +2\alpha' \nabla^2 \Phi -4 \alpha' \nabla_\sigma \Phi\nabla^\sigma \Phi +\frac{\alpha'}{6} H^{\lambda\rho\sigma}H_{\lambda\rho\sigma} \Big) \nonumber\\
=&\, \alpha' {R}^{\mu\nu} + 2\alpha' \nabla^\mu \nabla^\nu \Phi - \frac{\alpha'}{4} H^{\mu\rho\sigma}H^\nu_{\;\;\rho\sigma} -\frac{1}{2} G^{\mu\nu}\Big(\alpha' {R} + 4\alpha' \nabla^2\Phi - \frac{\alpha'}{12} H^{\lambda\rho\sigma}H_{\lambda\rho\sigma} \nonumber\\
&
- \frac{2(D-26)}{3} i -4 \alpha' \nabla_\sigma \Phi\nabla^\sigma \Phi \Big)
\end{align}
and we see that if we replace the covariant derivatives by partial derivatives we do indeed find what we were looking for.
