Under an infinitesimal diffeomorphism the Riemann metric changes by the Lie derivative
$$
\delta g\_{\mu\nu} = ({\mathcal L}\_\xi G)\_{\mu\nu}=\nabla\_\mu \xi\_\nu+\nabla\_\nu \xi\_\mu
$$
and under a change of metric the Levi-Civita Christoffel symbol changes by 
$$
\delta {\Gamma^{\alpha}}\_{\beta \mu}= \frac 12 g^{\alpha \lambda} (\nabla\_{\beta} \delta g\_{\lambda \mu}+ \nabla\_{\mu} \delta g\_{\beta \lambda}-
\nabla\_{\lambda} \delta g\_{\beta \mu}) $$
If one plugs the variation of the metric into the variation of the Christoffel sysmbol there are many terms, but in Bardeen and Zumino's famous paper on anomalies ((Nuc Phys B 244 (1984) 421) 
they assert (eq 4.10, but in my notation) that
$$
\delta {\Gamma^{\alpha}}\_{\beta \mu}=\xi^\tau \partial\_\tau {\Gamma^{\alpha}}\_{\beta \mu}+ \partial\_\beta \xi^\sigma {\Gamma^{\alpha}}\_{\sigma \mu}+ \partial\_\mu \xi^\sigma {\Gamma^{\alpha}}\_{\beta \sigma} - \partial\_\sigma \xi^\alpha {\Gamma^{\sigma}}\_{\beta \mu}-\partial^2\_{\beta\mu}\xi^\alpha
$$
This last equation makes sense as an infintesimal co-ordinate transformation or Lie derivative and greatly simplifies their later algebra. Is there a straightforward way to see the equivalence of the two computations? It's not plug and chug, but seems to lead into wasteland of Bianchi identities. 



Diffeomorphism variation of the Christoffel symbol
If you consult Sec 3.1 of Wald's book on GR, you will see how Christoffel symbols can be interpreted as *tensors* parametrizing the difference between the coordinate connection $\partial$ and the Levi-Civita connection $\nabla$, symbolically $\Gamma=\nabla-\partial$ (cf. Wald's (3.1.7)). You are interested in $\Gamma'=\nabla-\partial'$, where $\partial'$ is the new coordinate connection. Writing this in another way, $\Gamma' = (\nabla - \partial) + (\partial - \partial') = \Gamma + (\partial - \partial')$. Both terms in the last equality are tensors, so once you know them in one coordinate system, you know them in all coordinate systems, following the usual tensor transformation properties. The last formula in your question is essentially $\Gamma'$ written in the primed coordinates, to first order in the difference between the original and primed coordinates. The $\Gamma$-dependent terms are just the tensor transformation rule for $\Gamma$ interpreted as a tensor, while the $-\partial^2\xi$ term is precisely the $(\partial - \partial')$ difference. I'll leave checking that last claim as an exercise.


