I'm an actual linguist (it's my job, not a hobby), and I have never heard that except from wags, hacks, and the occassional conspiracy theorist. We know a great deal about old Gaelic languages, and they don't look at all like Norse languages. Verbs are conjugated differently, pronunciations are wholely different, dipthongs are substantially more abrupt (less rounded) in Norse (and later in Scottish Gaelic, as it absorbed this trait from Norwegian invaders). Additionally, even during the period of viking invasion, their affect on the Gaelic languages (except Manx) isn't really that profound (based in part by how different they are, and the lack of total immersion of them into one another); it certainly had a clear, definite effect, but it was not nearly so transformative as it could've been.
The Manx were recorded as speaking both their own language (a Gaelic language, which, at the time, was not unlike Irish), and had a Norwegian aristocracy for a while that spoke a Norse language, after the vikings invaded Mann. They were plainly set apart by language. Manx absorbed the most Nordic influence of all Gaelic languages, and it STILL isn't that similar to Norse, except in the blatant sense (that is, loan words, which are often extremely similar). In all three languages' cases, however, Latin and English had more of an effect (all of them experienced long periods of immersion in English, and all of them adopted aspects of Latin when Christianized). Early and Old Irish are not some mystery; we know what they looked like and how they were spoken (the sheer amount of literature written in the languages is staggering; Irish monks copied innumerable texts, mainly in Irish and Greek at first, then started using more Latin in the middle ages).
There are far too many differences between Old Irish and Old Norse (specifically western Norse) to think that they were that closely related. That all said, there was a Hiberno-Norse trade cant (a series of simple phrases and words) that was used by merchants in the region. It was not a regular language, and was just used to give the Norse and Gaels a bit of ease in trading. Conversely, the same cant fell out of use in the late 1300s or early 1400s, but it could hardly be considered a language (Rohn de Vella briefly mentions it in a dictionary of Britain, and notes that it seems to have had less than 300 words).
The assertation that they spoke the same, or near the same language is ludicrous; have you ever read anything in old Irish and compared it with old Norse? Consider some other problems with this theory; in Ireland, Norse lords had to learn Irish to effectively administrate. The Gaelic population had a rather heavy aversion to speaking Norse a lot of the time (it was considered the language of pagans for quite a while), and aside from that, it was difficult to learn. On the other side of things, the Norse had trouble learning Irish. There are numerous references to the horrendous butcher of the language by non-native speakers (which shouldn't be that unusual; the Gaelic languages are very complicated to learn, often, for non-native speakers, unless they have a pre-existing relation with the language and a basic understanding of it). If their languages were so similar, why did they trouble learning eachothers, and why did they even have to learn them? There were occassional Scottish chiefs in Ireland and vice versa in the period, and niether of them had any reported trouble administrating due to their language; their languages were genuinely very similar.
That all said; some similarities do exist. But those are generally pretty old (like the name of 'Loki' is clearly related to 'Lugh'; Loki used to mean 'The Blazing One', and Lugos (from Gaul) was 'The Shining One'). Most of the similarities far predate the period, and aren't really that intense. During and after the period, there are clear loan words and some additions to the language, emergence of a few dialects, etc., but that's entirely different. In Ireland, Scotland, and Mann, though, there were Norse-speaking minorities for a few centuries, but they all disappeared during the high/late middle ages, absorbed into the Gaelic-speaking populations. The most basic refutation is that Gaelic languages are Celtic in origin, and Scandinavian languages are Germanic, and the two families of languages are extremely different.
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