To date censuses have been taken in 1926, 1936, 1946, 1951, 1956, 1961, 1966, 1971, 1979 (the census due in 1976 was cancelled as an economy measure), 1981, 1986, 1991, 1996, 2002 and 2006. The returns for 1926 - 1946 and part of those for 1951 are held in the National Archives, but they remain under the control of the Central Statistics Office, to the extent that the staff of the National Archives are not permitted to examine them for any purpose. The more recent returns are still held by the Central Statistics Office. The 1926 Census Returns will be released to public inspection in January 2027.
The original census returns for 1861 and 1871 were destroyed shortly after the censuses were taken. Those for 1881 and 1891 were pulped during the First World War, probably because of the paper shortage. The returns for 1821, 1831, 1841 and 1851 were, apart from a few survivals, notably for a few counties for 1821 and 1831, destroyed in 1922 in the fire at the Public Record Office at the beginning of the Civil War."
From an additional source "The Four Courts building in Dublin housed the public records of Ireland. But in June of 1922 it was the scene of a battle between Free State forces and Republican Irregulars who had made the Four Courts their headquarters. The Free State bombardment set the building on fire midday on 30 June, and shortly thereafter a land mine exploded, fire spread, and the conflagration destroyed the building. Documents were found in the River Liffey, three miles away.
The flames consumed the original returns of the 1831 census. Records that can now be consulted come from the retrospectively amended records (1834) which incorporate a column for religious affiliation. One way or another, a number of early census records have survived. Best represented among the fragments are the counties of Cavan (1821), Galway (1821), Offaly/King's (1821), Londonderry (1831), and Meath (1821). Details appear in a variety of publications.
These returns have been microfilmed and are in the collections of the LDS Family History Library in Salt Lake City, Utah. For anyone interested in the descriptions written by the officials of the Public Record Office, refer to Irish and Scotch-Irish Ancestral Research (Falley, 1962) which quotes liberally from the Deputy Keepers reports,
primarily the 55th, 57th, and 58th, that were published after the 1922 fire.
Cemetery Records in Ireland
Parish records, of course, are a well-known and well-used source of information about Irish ancestors. There are also some searchable databases of Irish cemeteries by county available here: http://www.interment.net/ireland/index.htm
To search parish records for Irish ancestors, you may try this site, which claims to have largest collection of Parish records for the island of Ireland that are searchable online http://ifhf.brsgenealogy.com/
This website has an extensive list of links relating to Irish genealogy covering many aspects and facets which may help in your research http://www.genealogylinks.net/uk/ireland/