Why do Irish people pronounce 'th' as a hard 't'? Is that the original pronunciation of migrants from Britain which became corrupted in Britain by other languages perhaps? Or is the Irish pronunciation influenced by languages that were absorbed into their culture and not ours? Or the influence could have come with older languages like Old Irish or Gaelic that were more prevalent in Ireland than Britain. Anyone know?

Old Irish had the [θ] sound, but in Modern Irish it's become [h]. I'm not exactly sure of Scots Gaelic, but I believe it's the same as Irish in this respect, and has [h] for native th. So that lack might explain it; but we don't hear this pronunciation by Scots.

I don't know of any other pertinent influence on Irish English; the Norse of Dublin had [θ]. The Scots imported into Ulster wouldn't have done it, and in any case it's just as characteristic of Southern Ireland. Basically, sound changes just happen, and don't need an identifiable cause.

Note that three and tree are not normally homophones for the Irish, since the th remains dental, like a French or Spanish [t].

aput, what does it mean that the th remains dental?

English [t] is alveolar, with the tip of the tongue against the alveolar ridge just behind the teeth. In French and Spanish it's dental, with the tongue actually on the upper teeth. Both varieties are about equally common in the world's languages. A small number of languages have both (Australian and southern Indian languages, typically).

The [θ] sound of thick is dental too. In English you get a dental [t] when it comes before [θ] or the voiced [ð], as in eighth or hit them. As it only occurs in this context it goes unnoticed.

In Irish accents, for many speakers anyway, the dental [θ] has become dental [t], but isn't the same as the alveolar [t] they have in other words.