Most churches of all denominations in Ireland tended not to keep burial records- the exception is the Church of Ireland which usually did keep a burial register (although possibly other religions buried in COI ground may not always be recorded). People would mostly have had an idea where their family were buried and often the local sexton remembered exactly who was buried where.
Church records of marriages, baptisms, communicants, etc. vary from congregation to congregation. Also remember that not all records are online and may still only be in local custody.
Many Catholic parish register are online (free)-
https://registers.nli.ie/Sometimes there are also records on pay sites or in local custody which you can check for individual churches by clicking on 'Irish Ancestors' on the church's page.
Castletara parish, which marches Annagh, has online records well back to the 1700s-
https://registers.nli.ie/parishes/0822