If you give Iraq to Iran, they probably won't annex it, probably keep it as a puppet state... and then you'll have Iran and Syria together as it were.
So Iran won't stop at Iraq. It'll go and help out Syria against their ISIS group which is the main force of the revolution against the loyalists. And then Iran will have 2 puppet states.
It is not a wise idea to do that. If the USA entered Iraq to end Saddam's rule and bring democracy, do that. The job is clearly unfinished.