They're phagocytosed and broken down to components that can be recycled.

Viruses can be destroyed outside cells: Often by antibodies, but also by defenses like complement or innate immune lectins. In most cases these defenses physically stick to the outside of the virion and mark it for destruction by neutrophils, macrophages, or other specialized phagocytic cells. Phagocytosis means that the virion is taken up by these cells, and moved into extremely destructive little compartments that include acids and aggressive enzymes that break down proteins and nucleic acids down to their component parts. The resulting fragments can be recycled by the cell, or simply flushed out of the body.

Viruses can also be destroyed when they're infecting a cell. In this case the usual route is to destroy the infected cell, either by breaking it open or by forcing it into a specialized death pathway (apoptosis, or related pathways). In either case, the resulting cell debris is taken up by phagocytic cells and recycled or flushed.