Where does the Irish idiom "at all at all" come from?
To take your questions in reverse:
Where does the idiom come from?
As described here, in most English dialects at all stresses that you are considering a small amount, and in some Irish dialects it is used in two ways in questions:
To emphasise puzzlement over the answer if the question is one of fact. This is an extension of the more widely-found meaning (e.g. "How did you do that, at all?" can be interpreted as implying "I don't know how you could do even a little part of what you did", hence emphasising your puzzlement).
As a more general emphasis on any question, especially one of opinion. This stretches the meaning from the specific emphasis above to a more general emphasis. It can also serve as some added redundancy when the sentence is short, which is a verbal habit in quite a few dialects.
That answer was about questions, but the same also applies to statements in the inverse; at all emphasising "to the slightest extent" or else being generalised further to a more general emphasis—indeed the form in questions comes from the form in statements.
Now, there is also a tendency to double for emphasis in all forms of English that gets expressed in different ways. Even standard English would allow the doubling of some adverbs ("they were very very doubtful until he gave an example") and some adjectives ("big big" would be considered childish but certainly found while "great big" is equally redundant as while it uses different words, it uses them in synonymous senses) or redundant do which is semantically a doubling ("I do agree this isn't quite the same thing").
Hiberno-English is particularly fond of emphasis through redundancy, and it's not unheard of in Ulster-Scots either. Some of this comes from the way the reflexive would be used in Irish leading to a use of the reflexive for emphasis in English, some comes from the lack of a simple yes or no in Irish leading to either it not being used in Hiberno-English ("I am" instead of just "yes") or it being doubled with the standard yes, ("Yes, I am" or "I am, yes" instead of just "yes") which is truly redundant*, some from a different use of do-support for emphasis to standard English, ("I do be checking that every day.")†. None of these forms are originally emphatic, but having a knowledge of how they're redundant compared to the standard English equivalents meant that they're available for use for emphasis.
And then some grows from that; we've a form of English that has several features using redundancy for emphasis, and so we've a tendency to add more redundancy when we want to add emphasis, beyond that already found.
So, given the use of "at all" for emphasis, and given that tendency toward redundancy for emphasis, and given that doubling is found in all forms of English for emphasis, we end up with "at all, at all" for even more emphasis.
Do Irish-English speakers actually say this, or if they don't say it now, did they say it in the past?
Yes, and more so in the past.
Now, first it's worth considering that not all British people will say "Cor, blimey, Guv'nor!" nor all American people will say "dag nab it", but both of those expressions are stereotypical expressions of people from a particular background in a particular part of those countries that does get heard in real life.
Of the various expressions I give above, some I would use in all but the most formal registers (avoiding a plain "yes" or "no"), some I would never use, and some I'd only use quite consciously. Other Irish people from different areas would have a different set of Hibernicisms‡ and a different set of retentions of standard English that have since been lost (e.g. in much of England, they don't call presses, presses any more, having completely replaced the word with cupboard).
"At all, at all" is considered to be a bit stereotypical in much of the country, and so while you will definitely find some Irish people (especially older Irish people) using it as part of their everyday speech, you'll also find it used with a degree of irony, along with other "Oirish" phrases**.
Of course, use a phrase ironically often enough, and you end up just using it.
But that's about the balance I'd say "at all, at all" is at; some Irish people do say it, and some Irish people say it ironically.
*Once going through airport security I was asked if I had any electronics in my bag. I replied, "I don't, no". Unfortunately while any Irish person would be likely to interpret that as I said it, the security guy wasn't Irish and what they heard was "I don't know". This is not what you want a security guy hear you saying about your carry-on luggage.
†A stereotypical Hiberno-English turn of phrase is "I do be doing…", where it's not really particularly likely to be said over this use of do-support with any other verb††, but the double do makes it stand out more as strange to other English-speakers, hence it becoming a stereotype.
‡And for that matter, the dialect of my childhood also showed some Ulster-Scots influences, though I've lost at least some of them since.
***Oirish* being a slang term for stereotypical features of Irish culture, especially if they're not ever found in Ireland (e.g. most things involving leprechauns), are actually more a part of Irish-British or Irish-American culture if people falsely associate them with Ireland (e.g. the Irish-American fondness for corned beef or their feast-day "St. Patty's"), or are indeed found in Ireland, but not as much as stereotypes would suggest (ah, to be sure, to be sure, we do be having all manner o' funny things said at us that we don't be understandin' around the middle of March).
††Though I've only just realised something else about "I do be" which is that it is habitual be. I do be programming (I regularly program) but I am not programming (I am not currently programming). I only realised this feature of Hiberno-English "do be" when reading something about how AAVE similarly uses be in a habitual sense, "I be programming" etc.
Reduplication is an alleged trait of Hiberno-English (Irish-English), eg. to be sure, to be sure..., strongly associated with stage-Irish.
The OED has citations of this use from 1375.
The usage in negative constructions has a similarly long history, where it means ‘to any degree’.
I've read the old joke goes that in Dublin a yellow line on the road means no parking at all, and a double yellow means no parking at all, at all.