Iran has been very active in supporting the Shia Arab government in Iraq against ISIL (Islamic State in Iraq and the Levant), but not very public about it. This is because many of the things that ISIL is hated for (restrictions on women and on what people drink and do for entertainment) are the same things that have long been enforced in Iran. It is possible for Iran to condemn the ISIL tendency to slaughter lots of people just for being different (not Islamic or not Islamic enough) but they are reluctant to go into much detail, as least in the media. Iran would like ISIL to just go away, permanently and with great violence if necessary.
Meanwhile Iran has other religious problem in Iraq. Although Iran has considerable control over the Iraqi government, the same cannot be said for the Shia clergy in Iraq. While the Iranian Shia clergy have long believed in clerical control of the government, the Iraqi Shia clergy have not. This dispute became public in 2013 as Iranian Shia clergy issued fatwas (religious rulings) calling on Shia men to volunteer to fight for the pro-Iranian (and minority Shia) government in Syria (where the Sunni majority have been rebelling since 2011). The Iraqi Shia clergy were not nearly as united or enthusiastic about that sort of thing.
The Iraqi and Iranian clergy are also split over which of them is supreme in religious matters. While Iran has long had the most religious scholars and schools, Iraq has most of the key Shia shrines and a reputation for religious scholarship that was suppressed during several centuries of Sunni rule in Iraq. That ended with the overthrow of Saddam in 2003 and the senior Iraqi Shia clergy (many in exile in Iran) have been trying to assert their power and independence from Iran ever since. Thus the Iraqi Shia clergy do not encourage Shia men to go fight in Syria and assert that the war there is political, not religious. Thousands of Iraqi Shia men went to Syria to fight, although that came to halt by June when ISIL took Mosul.
Iran continues to ship weapons and other military supplies to Syria via Iraqi roads and air space. So far Iraq has resisted Western (especially American) and Arab pressure to halt these shipments. Part of this is because of Shia loyalty on the part of Iraqi politicians, but there is also the hefty Iranian bribes Iraqi politicians receive as well as the implicit promise of Iranian military assistance if the Gulf Arabs should attack Iraq in an attempt to restore a Sunni Arab dictatorship. That promise includes lots of military assistance against ISIL, but there is not a lot of Iranian publicity for this either.
Shia Arabs are having a harder and harder time accepting the Iranian argument that it is the duty of all Shia to fight Sunni Arab rebels in Syria in order to defend the Shia dictatorship (the Assad family) there. This bothers Shia Arabs because the fighting is Arab versus Arab for the benefit of non-Arab Iran. There are also problems with the Assads, who are secular dictators and most of the Iraqi Shia volunteers are Islamic conservatives. Worse, the Assads are Alawites, a Shia sect that is usually considered more heretical than Shia and only approved by the Iranian Shia clergy because Iran really needed an ally during its 1980s war with Iraq. Back then, Iraq and Syria were run by feuding branches of the Baath Party (a secular socialist political movement that was once very popular in the Arab world). Baath is still popular with Assad supporters in Syria but has been discredited most everywhere else. Iran is also mindful that Iraqi Shia fought bravely against Iran during the 1980s war, even though many of these same Shia later rebelled against Saddam. The basic truth in the region is that ethnicity (Arab versus Iranian) is stronger than religion (Shia versus Sunni).