Climate change jeopardises human immortality in two main ways. Firstly, the generally forecast changes are likely to result in water shortages, more extreme weather and mass migrations - all of which risk the breakdown of society which would prevent the continuing technological improvements to enable us to live forever. The second less likely, but more significant, event would be a rapid change in climate - remember the ice ages weren't that long ago. That would mean even if we had already attained biological immortality and had armies of civil engineer robots to build infrastructure we might not be able to fight the changes quickly enough and the whole human race would become extinct.
We're not going to keep up to date with every aspect of climate change research - but will track a few key topics.
Climate Change on MarsBBC - 11-Jan-2002
Severe episodic or catastrophic phenomena are much more common in nature than we have thought.
Sudden climate transitions during the QuaternaryProgress in Physical Geography - 01-Mar-1999
Some and possibly most large climate changes (involving, for example, a regional change in mean annual temperature of several degrees celsius) occurred at most on a timescale of a few centuries, sometimes decades, and perhaps even just a few years