To clarify something - 'The IRA' usually refers to the Provisional IRA in the modern Troubles. The Provisionals didn't receive weapons from the Soviets - their main sources were either old weapons from previous campaigns, or from US-based supporters, or from the Libyan government.

The group which did receive some weaponry from the Soviets was the Official IRA. The Official IRA and the Provisional IRA came into being as the Troubles started in August 1969, when the Irish Republican Army split into two factions. Prior to this time, the IRA had a Marxist-inclined leadership who thought of the conflict in terms of a class struggle against British imperialism; and were reluctant to get involved in a fight between ordinary Protestant and Catholic working class people. It was this that caused the split, between those who agreed with the leadership (and who became the Officials), and those who thought the IRA should be a sort of 'catholic defence force' (and who gravitated towards the Provisionals).

The Soviet weaponry (about 70 rifles with grenades and ammunition, I believe) was provided to the Official IRA in late 1972; the Official IRA declared a ceasefire against the British around this time, and their main contribution to the troubles after 1972 was to get involved with feuding with other Republican splinter groups, so the weaponry probably had little to no effect.

As to other effects of the Cold War on the troubles, likely little-to-none. Some of the actors in the conflict may have been partly motivated by communist or anti-communist ideologies inspired by the geopolitical situation of the time, but none of the major NATO or Warsaw Pact governments (excepting the UK, obviously) seems to have had any significant material involvement in the troubles. Occasionally a conspiracy theorist (including high-calibre conspiracy theorists like late British MP Enoch Powell) will allege that the assassinations of Louis Mountbatten or Airey Neave were at the behest of the CIA, or someone like Claire Sterling might accuse the KGB of aiding the Provisional IRA, but there hasn't been much evidence produced for either of these theories.