Does general relativity influence climate (vs Newtonian mechanics)
In short: No. Unless multi-million timescales are considered.

The reason we keep teaching Newtonian mechanics, is because it is a VERY accurate approximation of a more general theory (general relativity) within the regimes of speed and gravity acceleration found in everyday life. Therefore, all the relativistic corrections to Newtonian mechanics in the Earth system are negligible. Both, for the internal operations of the climate and for the evolution of its orbital parameters on timescales of hundreds of millennia or shorter.

One effect, would be the minor role that apsidal precession have on the change of distribution of incoming solar energy described by the Milankovic cycles. But that can arguably considered negligible. 

But perhaps the most important contribution would be the influence of relativistic corrections in Earth's orbit eccentricity variations on multi-million years timescales.

State-of-the-art orbital solutions for the long-term motion of the Earth like La2010, do use relativistic correction, however they are rather small. The way the corrections are included in the model are described by Saha &amp; Tremaine (1994), where they point that:


  General relativistic effects in planetary motion have fractional
  amplitude of order $\frac{k^2}{c^2 r} \sim 10^{-8}$ at $r = 1 \,  \textbf{AU}$.


Meaning that the correction correspond to a 0.000001% of the total variation of the orbital parameters. Unquestionably, even such small corrections have a cumulative effect over multi-million years timescales. Varadi et. at. (2003), compares the output of a model using relativistic corrections (solid line) with one not using them (dashed line), and the following figure show the results over 3 million years, where can be observed that significant differences arise after 300,000 years.



However, it is reasonable to consider that such small corrections would have been overcome at some point in the past by some other unaccounted effects, like large meteorite impacts (or fly-by), solar storms, etc. Remember that La2010 go back 50 million years and other models go back several billion years, when such events were rather common. So, I would still argue that relativistic corrections are not very significant. However, in a chaotic system like Earth's climate, you can never rule out that a negligible forcing could have had an important effect.

Of course the answer would be different (and the relative importance higher) if we consider a planet orbiting extremely close to a super-massive star. But, I'm assuming you are mostly considering planets similar to Earth in that regard.

There are also some claims (the Damhsa Theory) that gravitational waves might have play a role in the very long term climate evolution of Earth. However, I'm not capable to really assess the validity of such theory, but I have to admit I'm skeptical about it. I've posted a question about that in Physics SE, an there seem to be wide consensus that such effect would be exceedingly small to ever deserve any kind of consideration.