Iraq is a very insecure country for Americans. This is largely due to a rising case of anti-American sentiment in the region, the advent of terrorist organizations like ISIS and constant sectarian violence among factions scrambling for power against a weak government.
Americans can travel to Iraq, however, their capacity to do so is greatly limited through many red tapes. Your passport will need to be valid for 6 months after your travel dates, you will need an arrival sticker and a residency stamp and you will need a blood test for HIV and hepatitis.
Furthermore, the state department warns of a high risk of kidnapping and violent crimes and informs travelers that it has a limited ability to help them in case of an urgent need to evacuate. When your government claims an inability to assist you in a foreign land, you know you should by no means go there.