What are the specifics about the war in 451?
Who started it? Who is the conflict between?
What is the conflict? Who is the victor?
How long has it been going on?

HJH-
I don't think any specifics are offered. We are asked to fill in the gaps with our own worst imaginings- kind of like in Lord of the Flies, where the war is important but never fully revealed to us.
In the case of F451, I am inclined to think of it as the similar to the war in 1984. No one knows how long it's gone on, and no one is even quite sure who we're fighting anymore- but it's important that we're fighting "someone".

I think DanB is on to something when he compares the war in "F451" to the nebulous nature of the war in "1984". Maybe the specifics of the war don't matter, but the ongoing condition of war is what we have to deal with. In these novels of the lack of individual action and thought, we live in a generic world of generic violence and generic being. Perhaps only specific, individual activity, decision, and will can transcend it. We escape the void of the generic with the antidote of individual awareness and activity.

The war is simply a continuation of what happens to these people in everyday life although they don't realize it. Remember the "relatives" who are always fighting on the parlor walls over nothing? This is the idea I feel Bradbury is trying to convey. Over time people forget why they are fighting (especially if they are uninformed) but they keep fighting because they are to involve to back down. I get the sense that this war had been in the making for many years before it raised its ugly head.