When Maliki accused Turkey of intervening Iraq's domestic affairs, perhaps he was referring to Turkey-Kurdish relations with the Iraqi Kurdish leaders and Turkey's better relations with the Sunni leaders of Iraq.
These days, many political observers claim that the US left Iraq in the hands of Iran, and Iraq has become dominated by Iran. Yet I am skeptical about whether Iran would fully dominate Iraq.
First, Iran does not have enough technological and economical resources for such domination. Second, the Iraqi state system and military technology is totally an American-based one and is superior to Russian-based weapon systems. Therefore Iraq would not want to risk its advanced military system to go hand-in-hand with Iran.
Second, two other influential actors in the region, Saudi Arabia and Turkey, do not want Iran to dominate Iraq. Saudi Arabia especially does not want to see a pure Iranian domination. It is a fact that Iranian domination in Iraq would deeply affect the Saudi's Shiite community that dwells around their oil fields where Iran would play a destructive role once Iraq falls into the hands of Iran. Therefore the Saudis would use all their influence to prevent Iraq from becoming an Iranian-dominated territory.
Because of the delicate balance described above, I do not think that Iraqi Shiite politicians would open their arms to welcome Iranian domination over their country. However, Iran has a number of tools to use in Iraq, too.
Given the fact that Iranian intervention in Iraq is not limited to Iraq's borders, this has implications beyond Iraq. Iran aims to build a Shiite buffer as a fortress to protect its regime, and they will want to use all available tools to become an influential actor in Iraq.
The tools that Iran would use are indirect. First, Iran has already established a Shiite social network in Iraq that can be mobilized in favor of Iran once Iran needs it. Iran would use the Shiite network to boost pro-Iranian views among the people in the street. This is a well-known method that Iran has been using in its neighboring countries to create a warm climate toward the Iranian regime. For instance, the Turkish town I grew up in in the 1980s and 1990s was under Iran's influence; many people were invited to Iran and informed about Iran, and Iran has successfully created a pro-Iranian climate that one even could consider the area a virtual Iranian island within Turkey. I am sure it would be much easier to build such a link with Iraqi Shiite communities to establish virtual Iranian islands in Iraq.
The second tool Iran would use is the militant proxies in Iraq. According to Western sources, one of the best-known militant proxies is called “Asaib Ahl al-Haq” (League of the Righteous), often referred to as part of Special Groups. "These cells specialize in certain types of signature attacks, such as the employment of Iranian-produced weapons or components, including the roadside bombs referred to as explosively formed projectiles (EFP). The latter munitions are particularly significant because of their lethality; although they have been used in just 5 to 10 percent of roadside bombings, they account for 40 percent of US casualties” (Michael Knights, Iran's Ongoing Proxy War in Iraq, 2009).
Iran would use such militant proxies to terrorize and destabilize Iraq if it decides that the destabilization of Iraq would function as a buffer to protect the Iranian regime. In fact, as soon as the US withdrew, Iraq turned into a country of chaos. It is childish to think that Iran, one way or another, is not behind this. At this time, Iran is destabilizing Iraq because it does not want the Syrian regime to fall. If the Syrian regime falls, Iran thinks that Iraq is the last battleground it would fight its war in the Middle East. Therefore it is very likely that Iran, by using its active political network in Iraq, has been carrying out its clandestine operations into Iraqi territory.
All in all, it is less likely that Iran would exert its influence through a Shiite-ruled Iraqi state, but it certainly would do that by using its proxies within Iraq.