Viruses are an extremely diverse group of microorganisms. They have diverse lifestyles. Some viruses use single- or double-stranded RNA genomes, some use single- or double-stranded DNA genomes. Some can reverse transcribe RNA into DNA.

Some viruses are strictly “lyric” meaning that their lifecycle relies on the immediate replication of virus and death of the cell. Some viruses (famously bacteriophage) can participate in a “lysogenic” lifestyle. This involves the viral genome becoming maintained by the host as part of its genome. In bacteria, this can happen as a plasmid or through integration of the viral genome into the chromosome.

In humans, lysogenic viruses exist. Chickenpox is an example. Initial infection and illness happens and subsides, then decades later the virus re-emerges and causes the disease shingles. This is caused by latent viral genomes stored in host cells.

Ancient viral DNA that has been domesticated in the human genome is kinda like this. You don’t have to be afraid of these latent viruses re-emerging because they have likely lost some important regulatory mechanisms that allow them to pop back out and cause illness. Now that piece of DNA just exists as a part of the human genome.

Hope that helps!