The Shining

The Shining book came out in 1977. The Amityville book also came out in 1977 so it's highly unlikely that either book influenced the other. Also, the final night of being in the house is not documented because the Lutzes claim it was too frightening. So we don't know if anything with an axe was ever supposed to have occurred or not.

The Amityville movie came out on 1979, the Shining came out in 1980. Kubrick was probably completely buried in his work on The Shining, especially based on the notorious stories about his repeated takes and perfectionism, to even take notice of what was used in The Amityville Horror.

Additonally, in The Shining book Jack tries to break down the door with a roque mallet. I would guess that an axe was used in the film because it's more reasonable than a mallet, and also got done rather quickly based on Jack Nicholson's skills.

So short answer...no.

Could be, it was very similar.