In the latest episode of the What If? series by Randall Munroe and Henry Reich (previously at Neatorama), the question is: "If someone's DNA suddenly vanished, how long would that person last?" While the mitochondria is the powerhouse of the cell, it's the DNA that tells a cell what to do. You might think that our cells should know what to do by now, and they would continue because of inertia, but that won't go on indefinitely. Each of our body organs have a different task, some more time-sensitive than others.
As far as we know, this scenario has never happened. But to give an answer, they looked at a few scenarios in which our DNA becomes damaged and can't do its job properly. Those are mushroom poisoning, chemotherapy, and radiation poisoning. The consequences of each are pretty ghastly, even though they don't completely destroy the DNA in each cell. Without DNA, we would certainly be up a creek without a paddle.
PS: Commenters cited the case of Hisashi Ouchi, which is a tragic and gruesome read.
Commenting on Neatorama will earn you NeatoPoints!