Information about the war in Fahrenheit 451 is specifically left ambiguous to show how little people care about world events that don't directly affect their lives. Montag notes that there are bombers flying overhead every day, but there is no indication of where they are going.
A radio hummed somewhere. ". . . war may be declared any hour. This country stands ready to defend its--"
The firehouse trembled as a great flight of jet planes whistled a single note across the black morning sky.
(Bradbury, Fahrenheit 451, Google Books)
People are so shallow that they don't care about the other things happening in the world. This small hint that the war is coming is overshadowed by people's obsession with meaningless entertainment, and it appears that private citizens simply don't care enough to take any action. The only thing that is substantially known about the war is that it is ongoing, and that at the end of the book, bombers from the opposition destroy Montag's city and all the people there, including his wife Mildred. In this manner, Ray Bradbury satirizes public obsession with celebrity and pop culture, and the comfort most people take in what they believe to be an unchanging status quo.
There is no detailed information with regards to the war in Fahrenheit 451. However, there are several mentions of an eminent war throughout the book. Montag informs Mildred that they have started and won two atomic wars but does not give details as to whom they were fighting and for which reasons. Details of the war have intentionally been left out to depict the artificial lives of the people of that time. People have been brain washed and are pre occupied with meaningless things such as the TV families even as bombers transverse their skies daily. In fact Montag questions himself about how alien the subject of war is to them. He asks, “…Why doesn’t someone want to talk about it?...Is it because we’re having so much fun at home we’ve forgotten the world?....” After Montag escapes from the mechanical hound and joins the group of social outcasts who keep books alive, we learn that they are anticipating for the war to begin because it would advance their course by destroying the city and all its anti-book machinery.