- Statute law allows Ireland to revoke your naturalisation on the grounds of acquiring another citizenship.
- But this law has recently been found unconstitutional.
- In practice, the government does not do this anyway.
- In the case of the UK, it is particularly unlikely.
- As an Irish citizen, there is no particular reason to pursue UK naturalisation.
As of now, the possibility deprivation of Irish nationality for any reason is unclear, because of a ruling of the Supreme Court (Damache v Minister for Justice  IESC 63) that declared the statute allowing for this to be unconstitutional. That was because the law (section 19 of the Irish Nationality and Citizenship Act 1956) did not provide enough procedural safeguards. The Minister would initiate the process and make the final decision after expert advice, but was not an "independent and impartial decision-maker". It is now for the legislature to replace section 19 so as to cure the defect.
Under section 19, the grounds for revoking a certificate of naturalisation include:
(b) that the person to whom it was granted has, by any overt act, shown himself to have failed in his duty of fidelity to the nation and loyalty to the State
(e) that the person to whom it is granted has by any voluntary act, other than marriage or entry into a civil partnership, acquired another citizenship.
The issue in Damache was (b) following the appellant's conviction for terrorism offences in the United States. Regarding (e), it would seem on its face that acquiring UK nationality might trigger the possibility of revocation. However, in the scenario envisaged, that would leave you without any EU nationality, and in the Tjebbes case of 2019 (ECLI:EU:C:2019:189) the European Court of Justice found that such deprivation would only be possible after consideration of the specific consequences for the person concerned and their family. This point was not reached in Damache but would also tend to rule out any "automatic" loss of nationality; it would have to be the result of some longer and more involved process. Whatever replaces section 19 would have to be of this kind.
Special considerations also apply if loss of Irish nationality would leave you stateless, but that is not the issue in question.
Aside from cases of immigration fraud and terrorism, the State has rarely initiated processes under section 19. In fact, official guidance on immigration and nationality admits a policy of dual nationality being allowed.
For the United Kingdom, there are many people who are dual British and Irish nationals. The understanding between the governments with respect to the Common Travel Area, the Good Friday Agreement, and the general historic situation, would make it very unlikely that Ireland would treat acquisition of British nationality as a problem. The British-Irish Agreement of 1998 includes that both governments respect the "right to hold both British and Irish citizenship" for the people of Northern Ireland. Even if you are not a person of Northern Ireland, any action by the government of Ireland that would be seen as potentially touching on this right is politically untenable.
Irish citizens have full rights in the UK anyway
Under UK law, an Irish citizen can enter without a visa, live there as long as they want, get a job, claim benefits, vote in elections, be elected as an MP, and do everything that a UK citizen can normally do. There may be a vanishingly few exceptions for national security jobs.
Because of this, there isn't much reason why being naturalised in the UK would be worthwhile. You could do so for sentimental reasons but as far as the UK's concerned, you already hold a status that's just as good.
Extra note: UK honours
(In response to a comment below.)
The British Crown does grant honours, including knighthoods, to non-UK citizens and dual/multiple citizens, in some circumstances. It depends on whether the other nationality has King Charles III as head of state, and the attitude of the foreign government. Therefore, Canadians do not get knighthoods (Charles is King but the Canadian government would rather not), Belizians do (Charles is King and the government is fine with it), and Americans can get it as an honorary award that does not come with the Sir/Dame title.
For Ireland, Terry Wogan is an example of how this works. He was an Irish citizen who lived and worked in the UK for many years. In order to receive a British knighthood, he had to take up British citizenship (which was granted on an expedited basis), becoming Sir Terry. On the other hand, Bob Geldof is an Irish citizen (only) who holds an honorary British KBE, and is not formally entitled to be Sir Bob - much like American recipients such as Rudy Guiliani. The Irish government does not object to either possibility, although some individuals do.
An American-Irish-British triple citizen would be able to receive a UK knighthood and use the title Sir, assuming that the British government was willing to grant it, and unless there were some foreign legal blocker (e.g. the U.S. government does not allow federal officeholders to receive overseas decorations).