From 1898, local authorities became responsible for the maintenance of graveyards in their areas. Part of the job was to maintain burial registers recording age, cause of death, address and often next of kin. Few survive before the 1930s.
Other non-Church-owned cemeteries in Dublin and Belfast had beeen keeping such registers from much earlier.
Below is a list of what's currently online.
Glasnevin cemetery (€) has more than 1.5 million burial records for the greater Dublin area starting in 1828.
Dublin City Library and Archives has a complete guide to graveyards and their records in the Dublin area, as well as the burial registers for Clontarf, Drimnagh and Finglas cemeteries.
Buried in Fingal has burial registers from 32 cemeteries in what was north Co. Dublin.