What would happen in a cell if the DNA was destroyed completely or changed in any way?

If the DNA of a cell is destroyed completely the cell would die i.e. cease to exist. A change in the DNA (germ-line mutation) would change the structure and/or functions of proteins in the cell.

Explanation:

The central dogma theory indicates that DNA is replicated to produce daughter cells. These daughter cells are then transcribed into mRNA (micro RNA) which are in turn translated into structural and functional proteins.

If the DNA is totally destroyed, no mRNA is formed and by implication, such a cell has no proteins. Now proteins are practically the "godfather" molecules in any cell, they maintain the selectively permeable cell membrane's integrity, they form the structural component of most cellular organelles, they are essential for transport of essential/toxic substances in/out of the cell, etc.

Looking at these functions, cessation of protein production is equivalent to cessation of cell life (apoptosis if DNA destruction is not stimulated externally).

A mutation (change in DNA) changes which protein is produced. A practical indication of this is when a single point mutation in hemoglobin produces sickle cell anemia!!