So I'm with the consensus that this is generally an absolute disaster for everyone involved but particularly the Republic of Ireland. Thinking about how it would actually go down/the implications. I think you have roughly 2 scenarios.
A brief border war and an embarrassing climbdown
Republic forces start crossing the border, the British armed forces in Northern Ireland is roughly a peer to it ( and off course there are huge amounts of additional forces on GB proper and easily accessible forces that could be drawn from say deployments in Germany if needed), the RAF devastates the Republic forces as they cross and they pretty quickly hitt very well dug in British troops (Given much of NI was already pretty well militarised and troops there fairly well defended in case of attacks by the IRA). The Republic's government realises they are off their meds resignations and apologies happen and the war ends very quickly. The USA does not have to get involved and is very happy about this, probably sets back the peace process in Ireland by decades the Republic agrees to pay reparations to cover the cost of the invasion Unionist parties get a massive boost both among unionists and more unaligned groups, and Ireland finds itself somewhat politically on the naughty step when it comes to the US for at least a few years.
This is really what should happen with any realistic Republic government at the time, but then again I also can't figure out any realistic government would actually have started the war in the first place so you somehow end up with something a lot more militaristic who actually try to continue the war after they realise they're not just going to get waived in and greeted as liberators you get into
Short but very bloody conflict
In this case, the Irish keep going; they might manage to take a few border towns and then start to get pretty shredded. The US is a nightmare scenario domestically Reagan, providing actual military assets to the British against the Irish is going to hurt a lot, but if the Brits invoke article 5 and he does nothing it makes NATO look like it's not worth the paper it's written on so massively costs him abroad ( unlike Argentina they can't really claim the exception for colonial wars applies which to be honest was a bit of stretch even there ) plus of course the ussr will be milking this for everything it is worth.
My suspicion is the US effectively bribes the Brits to not invoke article 5, offering them diplomatic support to the hilt on both Ireland and Argentina, and a great big chunk of economic aid to help them through this 'turbulent period', while internationally trying to reframe this as not the Republic of Ireland V the UK but instead some sort of rouge (and presumably communist sponsored) cabal taking over a peaceful Western nation, and trying to destroy NATO and the Irish peace process with a pointless war ( does not particular matter whether this is true or not it's just the only narrative they can really do let them save face). In part due to this the US will absolutely want greater regime change more then just a few resignations at the top for this, and will almost certainly get it.
In the war itself the British take some casualties but the Republic's Armed Forces are pretty much annihilated, If the government still reesists and the Brits have to start heading into the Republic proper I suspect things get quite nasty with lots of potential for civilian casualties etc. My prresumption is they basically put together an armoured column and make a beeline for Dublin, sieze the different mechanisms of state and at that point anything short of ASB mind control will force a surrender. This is unlikely to impact the Falklands war at all ( amusingly possibly making their lives easier as the US is more willing to back the UK diplomatically just to avoid the British feeling they have to invoke article 5 over Ireland to reduce pressure on the home front), the bigger challenge the British for the Falklands was getting their forces there, which is greatly unaffected by this and they can handily manage with troops still on the home islands.
Longer term implications of the short bloody war scenario
The Unionist government in NI is at this stage very much not nice and not acting in good faith, and they have an excuse (and a US and UK government who to be honest or unlikely to overly care after the war), I expect you see de facto forced population movements of, some Catholics out of chunks of NI particularly any of the border towns that were taken ( effectively the Ulster government taking any excuse to label folk as collaborators and push them over the border).This combined with a rally around the flag effect for More neutral NI citizens, likely slows the tendency of increased support towards republicanism by a good few decades.
Once the new government has been established in Dublin I strongly assume they do everything they can to blame what happened on the IRA and similar groups ( so undemocratic paramilitary radicals), effectively to speed their re-entry back into polite society, this is the narrative the US will be very happy with, but it means the paramilitary side of republicanism in NI effectively get lumped into the bad guys in Cold War politics find their sources of funds cut off etc. longer term the impact of this probably depends on how restrained the UK government feels. Both Unionists and the Republicans in NI during the troubles were pretty vile, and if this means the Unionists get to let themselves loose while the Republicans have their hands tied behind their back, then you see even more de facto forced population movements and aa lot of impoverishment of Northern Irish Catholic businesses, power blocs et cetera. This probably eventually leads to renewed support for republicanism in parts of the wider world but they probably got a good decade or two we are effectively the unionists can, push their score up. If the British government show more restraint, and have the army stop the unionists from doing anything extreme there is effectively an opportunity here to start to push a peace process but one more firmly in line with the wishes of both the UK and the unionists, I can very much see things go either way and in part it probably depends on things like the scale of British casualties during the war, and for that matter UK domestic politics.
Thatcher gets an even bigger boost, although long-term identity that affects things too much just means she hits even bigger highs before a roughly as per OTL decline.
The actual terms on the Republic would be fairly light. Neither the UK or the US want it to suffer particularly. I suspect it gets very heavily demilitarised, has to accept some sort of international observers/peacekeeping force on the Irish side of the border (maybe US-led?) Until things calm down effectively to root out paramilitaries, but no UK troops sticking around, and while reparations are paid I expect this is arranged with a US loan on very generous terms or maybe just effectively writing off some debt the British owe the American. However the fairly heavy loss of life and quite possibly devastation of their capital is coming at the worst time (just from this should be starting to transition into the period of the Celtic Tiger) which might make Ireland miss the boat on its optimal time to develop and leave in a much worse situation long term,