I'm not well-informed as to the current state of affairs on the ground in Iraq, but there is an important general point that is not being made.
The USA and its allies invaded Iraq with the stated justification that the existing regime was a danger. As I wrote earlier, that is a reasonable justification. One can certainly argue whether the invasion was advisable, but I accept that it was justifiable.
However, whatever strategic interest the allies have in what happens in Iraq, the Iranians have more. Expecting the government of Iran to stand idly by while western countries attempt to fashion a new government there is not only unrealistic but unfair. Of course Iran is going to seek out allies among the factions struggling for influence, and of course it is going to support them, and, in a a situation of civil war, of course that support is going to involve arming them as well as funding them. It would be stupidly reckless of the Iranian government not to arm its allies in Iraq.
Now, if, as is alleged, Iranian-backed groups are fighting against US and British forces in Iraq, there is a problem that needs to be addressed. But pure outrage that Iran could seek to challenge the "rightful invaders" of Iraq will not do. The same logic that puts American and British guns in Iraq puts Iranian ones there too.
One could argue that Iran should refrain from intefering based on an ideal of absolute subservience to the U.N., which recognises the current Iraqi government. I challenge the commentators most hostile to Iran to make that argument with a straight face: I would be quite unable to do so myself.
Alternatively one could take an absolute imperialist line, and say that a Pax Americana is being imposed in Iraq, it will all be for the best, and everybody else better help, stay out of the way, or be squashed. I think that is the line that is taken in effect by the hawkish commentators, but I'm not sure they are really doing so consciously. Not that it would necessarily be a bad thing if such a peace could be imposed without local allies or compromises, but I am sure it is practically out of the question.
I would favour an acceptance that Iran has legitimate strategic interests in the internal struggles of Iraq, and a positive outcome is more likely to flow from some level of cooperation and compromise. Such an approach is made more difficult by the hysterical rhetoric that both parties have used against the other, as well as by the outstanding dispute over Iran's nuclear ambitions. At the very least, Iran will be seeking assurance that a powerful enemy is not being created on its border, either in the form of a hostile Iraqi administration, or an American base for further aggression. Neither seems likely to me, but governments of every country tend towards the paranoid in assessing the intentions of such unstable regimes.
If the USA does not make it absolutely clear that it has no intention of attacking Iran, then the natural assumption is that it does have such an intention. And that being the case, it would be an essential act of self-defense for Iran to attempt to prevent such an attack by keeping Iraq too unstable. And then, of course, that would be used to show that Iran is part of the problem and that regime change there would be desirable.
The only case in which it would make sense to make a fuss about Iran's interference in Iraq would be if it was insignificant. In that case it can be used to build up opinion for an attack on Iran, while forcing the interference to continue to escalate wouldn't matter because it's not significant anyway. If it is a major problem, the only way to stop it would be to not make a fuss about it, but to try to assure Iran that it is safe. Threats will not be successful, as the more Iran is threatened, the more incentive it has to keep Iraq unstable.
Iran could be expected to favour an outcome in Iraq of a stable representative government, provided it is confident that would not lead to an American leader saying "we have achieved what we set out to do in Iraq, we no longer need the army in Iraq, let's do the same now to Iran, after all it worked in Iraq."