Let's say that the Romans conquer and assimilate Ireland as they did Britain
What happened in Roman Britain was far from being an assimilation, and closer to a creolization.
What made a provincial society Roman was the presence of Roman law and civic conceptions (on imperial and municipal lines), eventually religion with Christianism.
It did existed in Britain, but was essentially a south-eastern thing, the rest of the province being more of a military frontier (not unlike Illyricum), with undevelloped social-institutional structures (compared to most Roman provinces, for exemple neighbouring Gaul)
People as Ambrosius Aurelianus/Riotomagus (whatever they were the same person, as I think, or not) clearly represented the romanisation of late imperial Britain up to still interacting with continental business; but provincial elites and subdividions remained largely the same than before the conquest : arguably Romans tended to fit in the shoes of previous situation, but you didn't have a deep transformation of British structures, the province remained largely a latifundar country/military frontier for most of its history.
or find a Irish king whom they help unite Ireland under the condition that the Irish kingdom has to pay a small tribute to the Roman government.
While Romans did enjoyed a dominant relationship with border peoples, trough clientelisation, trade, military recruitement; they generally frowned upon tentatives of unification of these peoples, as it would have meant a political mobilisation of their ressources as well as a possible threat. It's really unlikely they would not only accept, but support such tentative.
Now, you did have a Roman presence in Ireland historically : it was essentially a Romano-Britton commercial presence (on a semi-regular basis) as Roman goods and coins in the island does attest the existence of exchange and trade in southern and eastern Ireland, which most certainly represented a huge source for Roman geographic knowledge,
as in Ptolemy's work, which focuses on coastal settlements
.
These exchanges were, so it seems, accompanied by a seasonal army presence, as hinted by remains of fortifications in Drumanach, for exemple. It's hard to be certain about a clientele-like relationship in eastern Ireland as it existed in the German limes, but it's nothing implausible as Britain and Ireland never were two separate worlds : wars and political changes in Britain had to have consequences and continuity in Ireland as it did in the IVth century and in the VIth century.
The presence of coins and goods could be as much a hint of a pretty much attested trade, than subsides or insitutional exchanges with Irish chiefdoms. In any case, even if Rome as a state doesn't seem to have maintained on the long term relations with Hibernia (at least no longer past the IIIrd century), exchanges were still presents (Gaelic raiding in Britain is just another, violent, form of exchange) which explains, IMO, the relative easy christianisation of the island.
This article is excessively interesting on this matter.
So, there's nothing really preventing a deeper Roman conquest coming from this situation, except that there wouldn't be a real reason to do so : Britain was already a good conquest for what matter strategical purposes and relatively rich in its southern parts (especially grain, which was used in trans-limes trade).
Agricola, c. 100 AD, however, might have supported a client king's claims over an Irish chiefdom, in order to go along the traditional lines of stabilizing a region and "opening" it to Roman influence.
It's still possible, in the IInd century, to have other expeditions as such : Romans still made so in Caledonia during Septimus Severius's reign, while it was more "beating peripherical people into sumbission", than full-fledged conquest (in the case of Septimus, forcing Caledonian and Maetes to remain in their clientelized position.)
Could something akin to this have happened in Ireland?
Historically, Ireland seems to have been relatively calm on these manners. I'm not sure there's a good reason why : we don't know of any major roman operation, even the possibility of such, after Agricola nevertheless, but with a relatively small change, I could see some trouble-maker, managing or trying to gain local power or to unify some of the chiefdoms in Ireland.
Doing so, it would to disrupting the political/economical status quo, leading to inner troubles enough for Irish people to raid Britain earlier than IOTL, itself echoing with troubles in Roman Britain (either as consequences of raids, or the kinship between Brittons as Brigantes and Gaelic tribes). All of this, eventually leading to a Roman expedition to make the Irish chief remembering its place. It's a bit of a convoluted progression, but here we go.
I doubt, for aformentioned reasons, it would lead to a total conquest, or even a small one, with Ireland remaining a very peripherical region for what matter Romania. But, while not really likely, a Roman short-lived provincialisation (as in Roman Crimea, or Roman Armenia) is still doable : it would be, tough, a Britain on steroids regarding the general lesser structural developement, and the military nature of the occupation.
Fast forward a few centuries in the 400s when the last legions are pulled out of Britain and the remaining Romans are left to fend off the Germanic invasions on the island stands as a bastion of Roman civilization untouched by the barbarians.
A good part of IIIrd centuries raids (and later ones) weren't only Germanic but also Pictish and Gaelic. The increasing pressure of these in the Vth century is why Romano-Brittons searched to increase the already present foedi and laeti to fight them back. The main reason for Roman withdrawal, that said, is not that the pressure was too great, but that Roman armies in Britain (which tended to be over-militarized anyway) were called as reinforcement in Illyricum.
A Roman Ireland would be probably , as said, essentially a march, a military frontieer rather than a full-fledged province :
think along the lines of Roman presence north and east of Danube. If legions are still pulled out of Britain as IOTL, there's no reason why they wouldn't be out of Ireland, because there would be no point keeping them there.
