Biologist here. That’s not where I’d put the association. I’m going to simplify things a bit because this is going to be a long post, but hopefully this will help.

I think you’re thinking about the effects of heat on proteins. We can destabilize macromolecules by heating them, sometimes causing permanent damage. That takes a lot of heat, though. The planet would be long dead before we had to worry about the environmental temperature denaturing proteins.

Biologist related joke: How do you unscramble an egg?
Feed it to a chicken.

So, when the microscope was invented, people were startled to discover what were obviously living organisms in a drop of pond water. Discovery led to discovery, and we ended up learning about bacteria and the germ theory of disease. Some bacteria are really good (in that without them you would die), and some are really bad (like if you get infected you could die), but most are neutral. I do not believe there’s any direct connection between the global temperature and the rate of mutation in bacteria. That’s not to say there’s not a connection between climate change and disease rates. I’ll talk about that later.

So later on we discovered viruses. Viruses are much smaller than bacteria. While a bacterium is a cell with a whole physiology going on, a virus consists of a bit of genetic material bound in an envelope of proteins. The envelope will gain access to a cell and inject its genetic material, which the cell’s normal processes will start to work on, producing more viruses. Many people don’t consider viruses “alive.” I do, because I’m approaching it from a different direction. Again, the rate of mutation there isn’t really directly affected by a rise in global temperature, but there is a disease rate relationship.

We only discovered prions more recently. They’re even more simple than viruses. They’re really just a protein. We don’t understand them very well yet. The danger comes from the fact that prions can force other proteins to become copies of themselves. They can cause things like mad cow disease. They’re incredibly hardy and don’t mutate faster because of climate change. They’re very scary diseases with no known treatment and in some cases unknown transmission paths.

This one is a prion.

I think we can find correlations between disease in animals and climate change. Climate change forces animals into new environments where they can encounter diseases they’ve never faced before and to which they’re susceptible.

Even more important is the common root cause of industrialization and human expansion and environmental destruction. Again, forcing more animals together magnifies the impact of communicable diseases, and things like the use of antibiotics to make factory farming as profitable as possible has the inevitable consequences of making diseases that are harder to treat.