All surviving official Regimental records are held at – www.nationalarchives.gov.uk - and not in Ireland.
All surviving soldiers records are also held at - www.nationalarchives.gov.uk - and some may also be viewed, for a subscription, at:-
Unfortunately, in the case of World War One records, most of these have not survived but what remains are the Medal Index Cards. See - medal-index-cards-ww1.htm and cc cards
A Pension Record Card look-up can also be requested (for a fee) at - 2960-great-war-pension-record-cards-and-ledgers-deeper-understanding.html
If your relative remained in the British Army after the disbandment of the Royal Munster Fusiliers, see - army_pack.pdf
If he subsequently joined the Free State Army contact the Military Archives at
Cathal Brugha Barracks
The Royal Munster Fusiliers Association holds its own private database which was compiled from newspaper, diary and other sources and mainly covers WWI. This is an ongoing project and will, eventually, cover the pre-WWI era.