In what month did Johnny Cade die in The Outsiders?

The events of the story take place in the fall.

Most of the action in The Outsiders happens in a short period of time.  When the story begins, it is fall.  It is not much later that Johnny dies.

Pony is a thoughtful kid, and he notices things like sunrises and sunsets and the changing of the seasons.  We can tell from the clothes people are wearing that it is getting cooler but not that cold.  Pony also specifically mentions the season.

The pool was empty now in the fall, but the fountain was going merrily. Tall elm trees made the park shadowy and dark, and it would have been a good hangout, but we preferred our vacant lot, and the Shepard outfit liked the alleys down by the tracks, so the park was left to lovers and little kids. (Ch. 4)

There are a couple of other references to cold that seem to confirm that the story takes place around October.  We know that school is in session, but it is still warm enough to go around without a jacket.

Johnny and I stretched out on our backs and looked at the stars. I was freezing--- it was a cold night and all I had was that sweat shirt, but I could watch stars in sub-zero weather. I saw Johnny's cigarette glowing in the dark and wondered vaguely what it was like inside a burning ember... (Ch. 3)

Since he makes a comparison tos sub-zero weather, it is not that cold yet.  It is most likely just getting cooler because it is autumn.

Even though the exact month is not mentioned, it is clear that the story takes place over only a few days during fall sometime in the mid-sixties.  It begins when Johnny and Pony meet Cherry, and then they have to run away after killing Bob.  Not much longer after that they are hiding at the church when it catches fire, and shortly after that Johnny dies as a result of his injuries from the fire.