They filmed some scenes with Peeves in the first film, but he was cut before the film was released. The character wasn’t revived for the later films.
I found a few interviews that touch on the topic of Peeves.
First, there was an interview with BBC Norfolk with Chris Rankin, who played Percy Weasley, where he explains that they actually shot scenes with Peeves, but they were cut from the film:
What's the longest time you've ever spent shooting a scene?
The longest scene I've ever shot was on this first film. It was the scene with Peeves the Poltergeist. We shot it in Gloucester Cathedral cloisters.
Sadly it never made it to the final take although I hear Chris Columbus is going to be making a director's cut in which it will be reinstated.
This would have been Philosopher’s Stone, and he was played by Rik Mayall. I’m not aware of that director’s cut being released, or any of that footage being widely available.
So why were those scenes cut? There’s an interview with Chris Columbus, who directed the first two films, which makes it sound as if it was just
Apart from casting, what was the biggest challenge putting the film together? Was it the sheer scope of the thing?
[…] 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.
As other posters have said, Peeves didn’t play a particularly prominent role in the books: although he gets a few lines, he’s not critical to any plot points. If they were trying to shed material, you can see why Peeves would be one of the characters who gets cut.
Alternatively, there’s also an interview with Steve Kloves, who wrote all but one of the films, who offers his take on why Peeves was dropped in the final cut:
On a different note, what about some characters in the books who never appeared on screen? What ever happened to Peeves?
Peeves was always an issue. Chris Columbus was determined to put him in the first movie. I think there were even some technological problems with him initially, and [not] being satisfied with how he looked. He was always a bit tangential. I think [Argus] Filch, in a way, became that energy in the movies. And he was actually sort of beloved at a certain point, Filch. So I think to have Peeves, it would have felt like we were doubling up on that. It's not exactly the same but it's a bit like that. But he was a character we all loved.