According to BBC America, Christopher Columbus said he was unhappy with Peeves' look.
So somewhere there are outtakes from Harry Potter and the Sorcerer’s Stone with Rik Mayall as Peeves, and had he been left in, he could have remained Peeves throughout the rest of the series. Director Chris Columbus cut the scenes because they weren’t happy with the look of the mischievous poltergeist. But let’s not be too sad, there’s a coda to this that once again serves to prove what a loveable rogue the man was.
In this interview with Empire Magazine, Columbus implies that the length of the story and the film contributed to Peeves being cut.
You’d assume I would have been a complete basketcase but I think the biggest challenge of the first film was just treading carefully on what we were able to do and tearing the movie down to under three hours. I loved the book so much that it was extremely difficult to cut elements out. One of my favourite characters never made the film – Peeves, the annoying, sort of, mischievous poltergeist. Those sorts of things, there was just too much to film. Our first cut of the film was about 3 hours and 20 minutes. We tore it down when we first previewed it in Chicago to, I think, about 2h 40 and the issue with most parents was the length. The kids were like, "No, no, it's too short. We need more!" That's what we were constantly fighting for: kids who notoriously have a much shorter attention span wanted a four-hour film and parents were just like, "Please cut another half hour out of this." So it was a very fine line.
Peeves' involvement in the books wanes as the series progresses, and since the films already had to cull plotlines and characters from the books, it makes sense that Peeves would not be included in later movies.