What, if any, paleoclimate data can be derived from igneous rocks?
Stable isotope signatures in igneous rock (primarily $\delta$18O) can be affected by climate, but the effect is complicated and unreliable enough that I doubt it can really be used to say much about palaeoclimate. You might be interested in my friend Lara Owens' dissertation in which she considers the influence of climate on $\delta$18O in lava from Mt Erebus, Antarctica. Oxygen isotopes in Erebus precipitation are some of the lightest in the world, and she suggests that this water, after incorporation into a hydrothermal system whose rocks were later assimilated into magma, caused the anomalous $\delta$18O in some of the samples.

It would be interesting to see if there is a global geographic trend in similar anomalies, but I haven't heard of anyone trying that.

A more direct way that igneous rocks can record paleoclimate is through the effects that the atmosphere can have on pyroclastic deposits. Isopleth maps often can reveal the direction the wind was blowing during the eruption. Clast morphologies might even tell you the weather -- I'm thinking of accretionary lapilli, which form when an ash-rich eruption plume encounters a rain storm.

In conclusion, you might be able to glean a few data points here and there, but nothing comparable to the amazing records available in sedimentary rocks.

Edit to add: I just went to a lecture by Jay Quade about figuring out the timing of uplift in the Andes, and whether it came before or after the Atacama became hyperarid. Among many other geochemical tools, he used $\delta$2H in hydration waters on tephra, primarily to figure out palaeoaltitudes. He said he thinks the relationship between $\delta$2H and altitude has been fairly constant over the period of investigation (back to 50 mya).

Edit to add: I had forgotten to point out that subglacial eruption deposits are often distinctive (tuyas, hyaloclastites, pillow basalts). Wherever these deposits are found, you know a glacier or ice sheet was present and if you can date the rock then you know when there was a glacier there. Some recent work has suggested that you can even figure out the thickness of the glacier using the concentration of dissolved gasses in melt inclusions in the subglacially erupted lava flow.