Do you not mean the Royal Irish Rifles as The Royal Irish Regiment wasn't formed until 1993, if so you could try the Royal Ulster Rifles museum. The RUR were formed from the Rifles. Royal Ulster Rifles Museum
Edited to add;
Ignore this as I was getting mixed up, this might be a bit better,
Reply from: Alan Greveson
Date: Friday 16th September 2011 at 4:57 PM
There is no obvious surviving individual service record for Thomas Lee. An Army Medal Rolls Index card showed he qualified for the British War Medal and the Victory Medal. He served in the Royal irish Regiment. As he did not qualify for the 1914-15 Star for service overseas before December 31st 1915, he did not go abroad until after January 1st 1916. The 2nd Battalion Royal irish Regiment was a regular army battalion which had served in France since 14th August 1914. Therefore, Private Thomas Lee, of the Royal irish Regiment would have joined the Battalion in 1916 or later as part of a draft of reinforcements.
The Army was reorganised in February 1918 and some units were disbanded. For example, the 6th Battalion Royal irish Regiment was disbanded on February 9th 1918 and seven officers and 296 other ranks were posted to the 2nd Battalion at Hamel, northwest of Cambrai, at 1.30 p.m. that day.
"Soldiers Died in the Great War" (HMSO 1921) and the CWGC Debt of Honour both recorded Thomas Lee died on 21 March 1918. However, those records provide evidence only for the battalion he was with on the day he died, so it is not clear if he spent all his wartime service with the 2nd Battalion. The 2nd Battalion Royal irish Regiment, therefore, had been substantially altered in its composition as late as 9th February 1918 and before March 21st had a strength of 18 officers and 514 men.
In March 1918, the 2nd Battalion served with the 49th Infantry Brigade in the 16th Division.
The Division was responsible for defending the northern side of the Cologne Valley north of St. Quentin and was deployed along a ridge centred on Ronssoy. The 2nd Battalion Royal irish Regiment was in the forward sector of defence, based on the neighbouring village of Lempire. To their front was a network of machine gun emplacements. On the morning of March 21st 1918, when the German offensive began, there was a thick mist and the machine guns were ineffective. A heavy artillery bombardment caused severe losses on the right flank of 49 Brigade, exposing that flank when the enemy infantry advanced at about 9 a.m.. Within an hour and a half, the 49th Brigade had been virtually destroyed. The 2nd Battalion Royal irish Regiment recorded 78 men killed outright and by March 30th 1918, the Battalion had been reduced to 1 officer and 31 other ranks.
On the morning of the 21st March, as the right flank collapsed, the 2nd Battalion was exposed with the enemy surrounding them on three sides. Led by a Major Harrison, they refused to give up and fought from defensive position to defensive position until their ammunition ran out. They fought their way out to safety in the evening when the 6th Battalion Connaught Rangers counter-attacked.
The war diary of the 2nd Battalion for this period is held at the UK National Archives at Kew in Catalogue reference WO 95/1979 "2 Battalion Royal irish Regiment. 16 Division Date: 1916 1918".
There is a bit of confusion here.. The OP asked about the Royal Irish Irish regiment (disbanded 1922) which has no links, historical or otherwise, to the current Royal Irish Regiment (or indeed to the Royal Irish rifles, who post 1922 were renamed Royal Ulster rifles) There is a wiki page at Royal Irish Regiment (1684 which you might find helpful and I would suggest that an enquiry to the RUR museum might be your way forward.
BTW the Royal Irish Regiment once fought in battle against the Royal Regiment of Ireland... I will leave it to the historians on ARRSE to explain how this came to take place
The Imperial War museum has an impressive library. When you book explain what you want and they get the material ready for you. I was researching my local battalion and they had the war diary, an autobiography of the CO during WW1 and a host of bits and bobs I would have never found.
Thanks chaps. looks like a trip to Kew and the IWM is in order to fill in the missing links of my Great Grandfathers war record. Him and his son were in the same platoon and were killed within days of each other in May 1915.