The literal meaning of tuaisceart na hÉireann is “the north (i.e. northern part) of Ireland”. Tuaisceart Éireann is the name of the UK-governed political region, much as Dáil Éireann is the name of the lower house of the Oireachtas — without an article. I’m not sure why the article is not used in such governmental instances, since to my knowledge in similar grammatical circumstances the genitive form Éireann is accompanied by the feminine singular genitive article na (and H-prothesis). The nominative and dative forms Éire and Éirinn don’t take an article, though, so perhaps that influenced the governmental usage?
EDIT: Here’s what the 2012 Caighdeán Athbhreithnithe don Ghaeilge has to say (§3.1.4, page 60):
Fágtar an t-alt ar lár roimh Éireann i roinnt leaganacha oifigiúla, áfach (Dáil Éireann, Seanad Éireann, Iarnród Éireann, Bus Éireann, Raidió Teilifís Éireann), agus i roinnt seanleaganacha agus leaganacha fileata (plúr ban Éireann, bánchnoic Éireann).
The Graiméar Gaeilge na mBráithre Críostaí addresses this case in particular (§7.7, page 54):
Ní hionann an bhrí a bhaintear as an teideal polaitiúil Tuaisceart Éireann agus as an ngnáthabairt tuaisceart na hÉireann.
Am I right in saying the difference between 'Tuaisceart hÉireann' and 'Tuaisceart na hÉireann' is the same as the difference between 'North of Ireland' and 'Northern Ireland'? Maybe omitting the 'na' is just the politically correct version? Would make sense that the latter would have been taught in school considering the government's position on the north prior to the Good Friday Agreement.
Tuaisceart hÉireann to my knowledge is not used, but tuaisceart na hÉireann is used for “the north of Ireland”, and Tuaisceart Éireann is used for “Northern Ireland” — the last of these isn’t a literal translation, since tuaisceart is a noun rather than an adjective. As the quotes above explained, despite the genitive Éireann normally taking an article, an article isn’t used in some “official” terminology such as Dáil Éireann, in some poetic examples such as bánchnoic Éireann, or for the political jurisdiction Tuaisceart Éireann. If you have a source that conclusively identifies political correctness as the reason why the article is omitted, I’d like to learn about it — but note that a brief search of oireachtas.ie revealed the Acht um Aer-Loingseoireacht agus Aer-Iompar, 1936, in which Tuaisceart Éireann was used, decades before the Good Friday Agreement.
EDIT: Precedent for the lack of the article even goes back to the Annals of the Four Masters (see definition I (d) in the eDIL entry for taca), which has Tadhg ua Baoighill sonus ⁊ taccadh tuaiscirt Ereann in an entry for the year 1222; there are additional examples of tuaiscirt Ereann for other years, so it wasn’t a one-off mistake.
It isnt used for political reasons. Anglophone interpretation is that Tuaisceart na hEireann is literally "the North of Ireland". The semantics suggests that Tuaisceart Eireann, meaning Northern Ireland (of in English translation would change the word "weight"), denotes it does not belong to Ireland.
That is why the Irish in here see Tuaisceart na hEireann but were wrong.
Duolingo just took a politically safe stance on something the UK felt necessary to alter. Probably so they can maintain plausible deniability.
...it is totally Tuaisceart na hEireann, though.