There are many types of cells which will never again divide. Some of them may not need DNA to perform their function. Are there any cases where the DNA is discarded after a final differentiation?
I recommend looking into terminal differentiation.
As one extreme example of terminal differentiation, xylem cells that form plant vascular tissues are simply dead. This is a case where not only the genome but also physiologic activity in these cells is irrelevant to their function.
A more famous case is red blood cell enucleation, wherein the genome of mature cells is lost in cell division.
In other words, yes, this happens.