As a general principle, overlay approaches do not require the underlying NAV AID to be serviceable at the time of the approach. However, the FMS database must be checked against the paper chart and the FMS accuracy of the nav aid position established.
Approvals will also generally require regulator involvement. Regarding raw data, if it's available, it should be monitored. The primary position reference though is GPS Primary with a fixed RNP value (for example 0.3 nm) on the PROG page.
So if (for example) the XYZ VOR approach is selected in the database with the XYZ VOR notamed as unserviceable, the approach can indeed be flown in managed NAV with an approved RNP value (ie 0.3 nm).
If the NAV AID is unserviceable and dual GPS/dual FMS/FMS position difference errors occur, then the approach must be discontinued (IMC).