We show in a simple theoretical quasispecies model that the replication dynamics of hepatitis C virus and a related model-system, the bovine viral diarrhoea virus, result in an effective reduction of RNA templates in infected cells. Viral fitness does not translate directly into RNA sequence replication efficiency, and hence the abundance of the viral master sequences diminishes over time. Our results suggest that genes not involved in RNA replication accumulate mutations over time because they do not undergo selection during this phase. The selection of viral RNA occurs not only during replication but also during the ensuing stages of the viral life cycle: (i) envelopment of viral RNA and (ii) successful infection of other cells, which also requires functionality of non-replicative genes. In particular, viral fitness requires the ability of the genome to encode structural proteins which do not encounter selective pressure during RNA replication. We conclude by discussing the potential value of antiviral drugs which inhibit selection on parts of the viral genome.