Do quantum mechanical properties of particles, such as uncertainty, probability, tunneling, and so on, affect mutations in the genome?
Yes. (I mean, that's the answer to your question. You didn't ask for specifics.)
Proton transfer along the hydrogen bonds of DNA can lead to the creation of short-lived, but biologically relevant point mutations that can further lead to gene mutation and, potentially, cancer.
So that's proton tunneling at work in DNA.
This isn't really a new idea. I found one paper theorizing this from the 80's.