The virus contains RNA, not DNA. For a PCR test this RNA first has to be copied into DNA (with an enzyme called Reverse Transcriptase) so to detect viral RNA you need an RT-PCR test.
I suppose that RNA outside of the virus particle would be broken down immediately (RNA is very unstable, DNA is not). So I think it would be unlikely that the (RT)PCR test would pick up 'historic', no longer infective, genetic material of the virus as opposed to genetic material from infective virus particles.