Aims: Star formation in massive galaxies is quenched at some point during hierarchical mass assembly. To understand where and when the quenching processes takes place, we study the evolution of the total star formation rate per unit total halo mass (Σ(SFR)/M) in three different mass scales: low mass halos (field galaxies), groups, and clusters, up to a redshift z ≈ 1.6. Methods: We use deep far-infrared PACS data at 100 and 160 μm to accurately estimate the total star formation rate of the luminous infrared galaxy population of 9 clusters with mass ~1015 Msun, and 9 groups/poor clusters with mass ~5 × 1013 Msun. Estimates of the field Σ(SFR)/M are derived from the literature, by dividing the star formation rate density by the mean comoving matter density of the universe. Results: The field Σ(SFR)/M increases with redshift up to z ~ 1 and it is constant thereafter. The evolution of the Σ(SFR)/M - z relation in galaxy systems is much faster than in the field. Up to redshift z ~ 0.2, the field has a higher Σ(SFR)/M than galaxy groups and galaxy clusters. At higher redshifts, galaxy groups and the field have similar Σ(SFR)/M, while massive clusters have significantly lower Σ(SFR)/M than both groups and the field. There is a hint of a reversal of the SFR activity vs. environment at z ~ 1.6, where the group Σ(SFR)/M lies above the field Σ(SFR)/M - z relation. We discuss possible interpretations of our results in terms of the processes of downsizing, and star-formation quenching. Herschel is an ESA space observatory with science instruments provided by European-led Principal Investigator consortia and with important participation from NASA.