A few questions to consider,
Would the Carolingians have emerged as a dynasty/empire in a pagan Western Europe?
They pretty much did that, didn't they?
Whether butterflies will prevent the arnulfings from becoming the dominant family in the Late Meroviongianstate is open to debate, of course, but there is no time at which Ireland being Christian is decisive for the success of the early members of the dynasty. The earliest point where you could make a case (indirectly) is the coronation of Pepin. With less emphasis on papal authority - which to a large degree was a product of the Anglo-Saxon church - that might not have happened, or at east not the way it did.
...would classical heritage have been lost with no "Irish saving Europe". Would Justinian have had a less troubled history with ihis leadership of Christendom unchallenged?
Classical heritage would continue just fine in the scriptories at places like Bobbio, Vivarium, or Cassino, not to mention Constantinople, Egypt and Syria. what might not happen is the Carolingian reform movement. That would have huge repercussions for Europem, but not for classical heritage. Justinian probably couldn't care öless what went on in Ireland.During his lifetime, the Irish monasteries were just being founded and we know almost zilch of what was actually goinbg on, for that matter. His problem was based much closer to home and far more urgent - the Irish, at least, were more or less orthodox, if barbaric, and had no way of ever becoming an issue in his realm.
Possible outcomes:
No European imperialism
Near East becomes the center of civilization. Islam empire would be uncontested and possibly not see the need to resort to extremism.
Nice though that would be, it's not likely given how extremist ancient Christianity and early Islam occasionally managed to be. The near east invented holy war (they should have patented it, they'd be rich on royalties by now). European imperialism - well, possible, but it's more likely IMO that European power projection would simply take on a different form.
A barbaric Europe means a less developed Europe in contrast to Moorish Spain and the Middle East meaning it leaves itself open for conquest.
It really doesn't work that way at the time. The military resources of the European kingdoms were considerable, and their kings could effectively deploy them against outside foes. They also did not depend on the ability to sustain a complex civilian economy to the same degree. Oversimplified, but basically: the Byzantines and Arabs neededrevenue so they could have an army, so they had tax collectors. The Merovingians and Lombards needed an army so they could have revenues, so they kept armed men. The system predates the reformatio imperii, it's not Irish or Catholic and there's no need to think it would cease to operate.
Of course the cohesion of the imperial state is another matter, but I doubt that it was that important for the military defense of the realm given that Europe's most successful defenses in the tenth century were fairly consistently mounted not by the imperial centre, but by local powers.
With no Charles Martel Gaul is conquered by the Moors in
the 8th century.
That is very much open to doubt, to say the least. Also, Charles Martel dpoes not depend on Irish support. Or even Church support, given how many bishops hated his guts. So even assuming he's the sole reason for Francia not becoming Muslim, he'd as likely as not still be there.
Viking raiders will still explode out of Scandinavia as a result of over population at home as in otl, raiding England and Ireland. They'll still trade with the Muslims as in otl and sell European slaves to the Muslims and Byzantines.
It's also quite likely that they'll be Christianised and become good Europeans at some point. The attraction was still there. they might give a non-Christian Ireland a wider berth, though - not so much worth plundering, and they fight harder to keep it.
Outside Europe, No Christendom means no Columbus and the opening of the trans-atlantic trade. That means the traditional overland silk roads will remain open and the pre 1492 world system trade won't suffer.
Wait, how did we get here from there? You need to read less Cahill, methinks.
China will continue to keep trading. Eventually some lost Chinese ship that gets separated from the treasure fleet accidentally discovers Mexico and instead of the Columbian exchange, TTL's discovery will result in a Zheng He exchange or whoever the lost Chinese sea captain happens to blunder his way into the new world. Chinese guns, germs and steel wipe out the indigenous Aztecs or most of them and Chinese/Monols reintroduce the horse into the new world.
Treasure fleets are a very different thing from Chinese trade. That said, a Chinese discovery is possible, the transatlantic route is more likely, though.