Why don't plants get sunburned?
The following study shows evidence that UVB does damage plant DNA.


Rousseaux et al. Ozone depletion and UVB radiation: Impact on plant DNA damage in southern South America. Proc Natl Acad Sci U S A. 1999 December 21; 96(26): 15310–15315.
http://www.ncbi.nlm.nih.gov/pmc/articles/PMC24816/