DNA is the instruction set for protein synthesis. If your DNA is damaged beyond repair that's an issue, and sure apoptosis is the usual result and not cancer. At the same time however when cells die either from the direct impact of DNA damage or apoptosis they both stop working and spill their contents into the area... including signalling molecules that promote inflammation.

The sunburn you feel isn't a distress signal from cells warning that they can't be trusted, it's tissue death and an inflammatory reaction triggered by a cytokine response.