Obviously, it is conceivable that DNA could be so damaged that the organism dies. Trivially, blasting cells with radiation will cause rampant DNA damage, which will trigger p53 and lead to the cells suiciding (apoptosis), this is a "deliberate" (so to speak) mechanism to manage cancer risk.
In practice, the reason DNA damage leads to death is usually not because the information is corrupted and important functions are abrogated, but because the mutations lead to cancer, and the cancer leads to death.
However, most people don't die of cancer. A lot do, especially in developed countries, but most often people die because of infection or heart failure. Other major causes include neurodegenerative disease (neurons dying) and diabetes.
If you are asking whether in some part of the DNA, there is a bit saying
Maximum_Age = 85 (by analogy to programming constants), I don't think anything of the sort has been found. The body naturally has finite lifespan because things break, this is extended up to a point because the body also has mechanisms to repair some kinds of damage.
Why not repair all kinds of damage? Why are the repair processes imperfect, and cannot maintain the organism indefinitely? Good questions, without good answers known to science.