The movie does not make it clear what became of the planet's moon. And the omission is intentional.
Despite the fact the moon was not visible in the sky after Heston's landing, it would not necessarily imply anything to him other than he could not see the moon at that time. Considering how chaotic the landing was, they might not have been aware of the moon one way or the other.
One possibility listed at the time of the movie's release was that it was blown up and formed a ring of smaller rocks and dust around the planet. That would account for one of the astronauts saying "There's always a 'strange cloud cover at night.'"
Blowing up the moon is no mean feat and would have likely changed the face of the Earth in any number of less-than-pleasant ways. I believe this was one of the ideas for the story to explain the state of the world (extreme desertification and the like) but was never covered in the dialog.
For at least three days a month the moon is not visible at all. Depending on the weather, it could have been at least another week before the moon became visible. Despite the retreat of the moon from the Earth, (3.8 cm per year) it would not have been significantly different-appearing in the sky.
Given they were astronauts and likely highly familiar with the appearance of Earth's moon, Heston would have known exactly where he was, even if there had been additional meteor strikes during the time he was traveling in time. Such strikes would not have made significant changes in the appearance of the moon to one who was aware of its general features.
The shock value of the movie is the big reveal at the end of the Statue of Liberty, it was better if the moon didn't give away where he was. So it was easiest to remove the moon from the story and leave everyone guessing.