So before I start getting into the Irish and Irish-German relations, there is something that needs to be cleared up:

This

I ask because I found a fringe group of twitter Nazi's that claim to be white even though they are of Irish heritage.

does not make sense in the European continental context that was so basic for the Nazis and it is even debatable if it makes sense in a US context, seeing as to how some scholars argue that the Irish in the US fit into an ideological metric of "whiteness" albeit as whites of lesser worth. And even in the UK, while it is put forth that Ireland was a testing ground for colonial policies, the issue is complicated by secterianism.

But that is all a bit besides the point, which is that for the original, German Nazis "whiteness" as it arose and had been around as a concept in the Anglo and specifically US context was not a leading or important category in terms of ideological thinking. What I mean is that while today in current political development in the United States, Neo-Nazism is closely tied with what can be generally described as "white supremacy". But that is build on an understanding of whiteness that is specific to the US context in the sense that "caucasian" or "white" functions as an alleged overarching racial category encompassing people who are essentially European, specifically in a society deeply impacted by its past of a slavery system right at home, in contrast to people of African, Caribbean and Latin American heritage.

This is not how it worked, how it couldn't work really, in a society like the German Empire and subsequently Weimar and Nazi Germany. Their racism was emphatically not based upon a category of "whiteness" like it was in the US because it made sense within the historical formations of society, state building and ideologically development. Yes, continental European were colonial nations – even Germany – and thus a colonial metric of alleged black inferiority was ideologically present but the model of racism the Nazis embraced – the völkisch racism – embraced a very different model that did not have "whiteness" in its center.

The racialist idea of Aryans and its other, the Jews, is build upon the idea of people being superior and inferior because of their "race", their "blood" but what defined these "races" was not whiteness in the US sense because this ideology never served the purpose of being weaponized against people like African slaves brought to a society with the specific purpose of being property, of being subjugated for economic exploitation on a massive scale. Nazi anti-Semitism and racial ideology rested upon the ostensibly contradictory idea that Jews were both inferior and at the same time conspiratorially powerful. For the völkisch ideologues, Jews were both weak yet in control of communism, the Catholic Church, and in general everything that fit their most important metric of the "international" (vs. the national, which was identified with Aryans and so forth).

Furthermore, whiteness played little role for the Nazi because instead of the idea of becoming a colonial power subjugating non-European peoples, their whole project was the superiority of the Aryans (Germans) in Europe, with complicated made-up racial theories creating hierarchies among people classically understood in an Anglo context as "white" (the French, Dutch and so forth) and people understood maybe as only half-civilized but fitting a certain metric of "whiteness" in the Anglo sense nonetheless (Slavs, Spanish, Greek).

In short, the idea that the Irish weren't white was not one that the original German Nazis would have understood. In their metric, the Irish would have been understood as "Westisch" or Celtic, which slated them on par with something like the French or even the Scots. Though I need to caution that whatever racial theory the Nazis formulated were all made up and thus rather open to be flexible and pragmatic interpretation except when it came to Jews.

So, what did this mean for Irish-Nazi German relations? Recent discussions within historiography have pointed out some facts that are seen as rather uncomfortable. Irish neutrality was kept during the war and the run up to it but certain political forces within Ireland saw the Germans between 1933 and 1945 as convenient allies against the British and vice versa.

Charles Bewley, Irish minister to Germany from 1933 forward, made several pro-Nazi comments, praising Hitler as the safeguard against communism in Europe. Similarly, Joseph P Walshe, Ireland's foreign minister made some rather enthusiastic comments about Nazi Germany in 1933. While the Irish political elites soon turned less enthusiastic when Nazi Germany in the first half of the 1930s denounced Irish Republican politics as part of their attempt to garner goodwill with the British establishment, the second half of the 1930s and into the 1940s saw some tacit collaboration between the Irish Republican Army and the Nazi German Abwehr as Mark M. Hull points out in his book Irish Secrets. German Espionage in Wartime Ireland 1939-1945.

Basically, the Abwehr had sent agents to Ireland in order to coordinate IRA attacks with the German attack on France in 1939 and 1940 so that British troops would be bound at home and disorder would be sown in Britain during an important time. This happened sometimes with the knowledge of IRA leaders involved, sometimes without. One IRA leader, Seamus O'Donovan, even visited Nazi Germany several times holding discussions with the Abwehr in order to initiate IRA attacks should a war break out. Indeed, one such attacked occurred just a few days before the war broke out: On August 25, 1939 the IRA targeted several British cities killing 5 and injuring 70.

Nevertheless and all in all, the cooperation proved ultimately pretty useless to both sides and not much ever came of it.

As for Irish citizens in Nazi-occupied Europe: They experienced the same restrictions that nationals from other neutral nations did but were never targeted or especially discriminated against by the Nazis. The same also goes for American or British POWs of Irish / North Irish descent.

Germans and a certain number of Irish far-rightists fought together in the side of Franco in the Spanish Civil War in form of the Irish Brigade of the Blueshirts and a tiny number of those Irish far-rightists also joined the Wehrmacht as volunteers later to fight on Germany's side. But those numbers were so minuscule that we have hardly any information on them. Terence O'Reilly wrote a book about them called Hitler's Irishmen in 2008 but since I have not read it, I can not comment on its quality.

Anyways, in summary, Irish were not discriminated against by Nazi Germany because Nazi racial ideology differed greatly from the idea of Anglo "whiteness".