Bacterophage
Its not something anyone ever really considers.. =P Good question! I assume the host DNA just floats down and interacts with other molecules and eventually degrades into its constituent parts. The remains are likely taken up by other bacteria and used again.
A virus injects its DNA into the host cell making it produce multiple copies of that DNA and multiple copies of the protein capsule of that virus. After a while, the host cell becomes full of many copies of that virus, then the host cell explodes releasing all the new viruses. If the host cell is a bacterium and the the virus is a bacterophage, this phenomenon is done in two ways either by the lytic cycle in which virus DNA survives and the bacterial cell is destroyed,or the lysogenic cycle in which virus DNA is incorporated in the host cell DNA.