Is Iraq a theocracy?

No Iraq is not a theocracy at the moment.
Iraq, since before the formal naming of the Iraqi nation as Iraq and before many of the religions themselves, had been a civilized nation with a strong culture for millenniums. Religions were gradually embraced upon their arrivals with no particular revolutionary civil bloodshed or power grabs (in general) as was the case for many other nations. The exeptions of the pseudo theocracies of the Abbassids, Ottomans and Buyyids and similar caliphate empires are difficult to explain, but we will keep that aside for now.

Eventually Christianity and Islam were the main major remnants of followed religions in Iraq, (with Christians currently fleeing to other Christian majority Western nations with better economies and less impact from wars due to modern ease of international transportation). However since Independence, no theocracy existed. A brief monarchy followed by a military autocracy did, however. This was, despite not being "democratic", one of the finest times for Iraq and made it one of the richest nations in the world with, in line with its thousands of years of tradition, a highly civilized and beautiful country and culture.

Iraq has ranged in its type of government over the past century, including Pan-Arab Nationalism and Secularism, however, a theocracy has never been Iraq's profile. Not to say that theocracy was necessarily always bad, but it wasn't recently the form of government there.

Despite the existence of religious holy sites in Iraq, and a correspondingly culturally-conservative (yet extremely tolerant) culture, the Political-Islamic agendas notable in other Muslim majority countries do not exist locally due to the different Islamic School of thought adhered to by Iraqis than the contemporary Wahhabi-salafist extremist Ideology that is currently gaining popularity. Wahhabist school of thought was rejected historically by the majority of prominent Islamic schools until this day, yet is recently gaining popularity by petro dollars in other places such as Egypt and also due to historic hegemony of Western and previous Imperialism on the alternative of regional Arab Nationalism. This extremism is still widely frowned upon in Iraq. Western intervention in the region has often experienced difficulty (or appeared to experience difficulty) in differentiating between anti-occupational sentiment and their arch enemy of religious extremism, and generally upset the fragile balance of the region and make a huge mess that will eventually have worldwide reaching implications.

Even though one of the major school of thoughts followed, the Twelver Jaafari Jurisprudence, has showed potential to develop theocratic style political interpretation such as in Iran, such a model was not popular in Iraq and previously refused by many of the followers of that particular branch in Iraqi politics, such as the current Prime Minister.

The current constitution tries to ensure that all ethnicities of Iraqis of all the different religious backgrounds get represented in the central civil government, at the same time with the efforts of doing that through democratic elections. At the same time they are members of the Arab league and have a majority Arab population, a population who, as a whole, historically feel culturally threatened by Persians, Western Imperialism, Kurdish Separatists and Jewish Zionist Invasions.

Iraqis, however tough they may come across as, are kind, simple folk that can unfortunately sometimes be also easily manipulated and steered in particular directions Ideologically. So future outcomes can be difficult to predict.