While in danger of losing Syria, Iran still exerts a lot of control over Iraq. Iraqi Arabs dislike the Iranians (who are Indo-European, not Arab and long-time oppressors of the Arabs) and the Shia majority government in Iraq knows that most of the Sunni Arab governments in the region would prefer that the Sunni Arab minority was again running Iraq (as it has done for centuries until overthrown in 2003). This is part of the great Sunni/Shia religious dispute that has been going on for over a thousand years. The Shia have seen better days, but now see a chance to gain more power because of the Shia clergy that run the religious dictatorship in Iran. The Shia politicians in Iraq are willing to do favors for Iran (like allowing aid for the Syrian dictatorship to pass through), just in case.
Iraq's leaders aren't controlled by Iran. They are hedging their bets in a dangerous region where the Sunni Arab world sees Arab-dominated Iraq as more Shia than Arab. And right now, Iraq's military is not equipped or trained for conventional combat against even a weakened Iranian conventional military.
Remember, Shia Arab soldiers in Saddam's army clearly felt more Arab than Shia as they fought Iran for eight years in the Iran-Iraq War. Once Iraq's armed forces are more capable of conventional warfare, Iraq's fear of Iran will diminish and allow Iraq to ignore Iranian wishes much more.
I was frustrated during the Iraq War insurgency that the Arab world sympathized with and aided the Sunni resistance and terror campaigns in Iraq rather than work with us to establish a stable Shia-dominated Iraq which could serve as a reliable counter-weight to Iran (and one that didn't drop dissidents in plastic shredders). I'd hoped that time and concern over Iran's nuclear ambitions with change this Sunni Arab attitude toward Iraq.
But then again, I hoped that we'd still have 25,000 troops in Iraq. That would have made Sunni Arab suspcicions of less importance to Iraq and denied Iran opportunities to weild influence.
Smart diplomacy alongside smart religious bigotry has worked out swell.
UPDATE: I meant to link my older post on the Iran-Iraq conventional balance. Silly me, I assumed we'd stay to defend what we bled to gain.