You'd get something very similar to radiation sickness. In radiation sickness, the cells most heavily affected are the cells that divide the most often since damaged DNA most often causes problems when the cell tries to divide. Typically the cell's machinery detects a problem, such as a heavily fragmented chromosome, and signals for the cell to cease dividing. If the cell can't correct the damage, it will either enter senescence, where the cell no longer divides, or it will try to kill itself via apoptosis.
Most of the symptoms of radiation sickness come from the death of large numbers of these cells and/or the inability to replace them. For example, the lining of the gastrointestinal tract normally undergoes constant cell division since the cells are exposed to such a harsh environment. Radiation damage kills off many of these cells and many of the remaining cells can't divide, leading to the eventual death of large sections of the gastrointestinal tract and the accompanying symptoms (nausea, vomiting, diarrhea, etc).
Since your superpower isn't outright killing cells the symptoms wouldn't be quite as severe as a large dose of radiation, but would still be lethal in perhaps a few weeks. Other injuries or sickness could drastically shorten this however. Certain infections can kill you in less than a day if your immune system is heavily compromised. Heavy bleeding from a physical injury would be even more life threatening than usual, as the body would not be able to produce replacement red blood cells. Nor would it be able to repair the injury. At best a clot would form, but no scar tissue, so damage would just accumulate over time.
I'd say that under the best conditions you'd be able to survive maybe a few weeks. The lowest radiation doses that produce acute radiation syndrome (ARS) typically kill in about 6-8 weeks, whereas extremely high doses of radiation can kill in less than 2 days. I'd split the difference and say perhaps 2-4 weeks at best?