Here's a question: what's a safe war? Prince Harry can't go to Iraq because they can't guarantee his safety. But we're talking about a war zone, which by my understanding are always dangerous. And, of course, Prince Andrew flew around in the South Atlantic during the Falklands War, which was hardly hazard-free.
The army decision is, of course, based on a different understanding. Iraq isn't like the Falklands at all. There aren't two national armies, signed up to international conventions on not being too nasty to prisoners etc. Terrorism is part of the way the war is fought, and should some militias capture Harry, we would be treated to some ritual humiliation or worse on the Internet. The PR and morale impact would be much greater than the sense of loss there would have been if Prince Andrew had been shot down, or captured as a POW during a good old-fashioned war like the Falklands.
It's a scary new world we live in, and a very scary new world our soldiers are required to fight in, and they all must be thinking about what might happen to them, whether or not they have a Royal in the tank with them. It's probably a correct decision, especially if it reduces the specific focus on other soldiers too, but it shows the desperate need to bring this conflict to an end somehow.