On my AT thruhikes, people built fires in the early days, when it was cold, but by Virginia they were pretty rare. As others have said, at the end of a long day, the hikers were too tired to go scavenging and generally the areas around the shelters had been picked clean by May. Occasionally someone would insist on a fire, even though it was too hot to be welcome, but I can only think of a couple of times that that happened - mostly in New England.
On the PCT and CDT campfires were even rarer. After hiking though miles of burnt forest, the idea of building a fire that isn't necessary for warmth really doesn't appeal. We built one fire on the PCT on a snowy day in a car campground in Washington, but I think that was the only one.
We got out of the habit after that, and rarely have a fire any more, even when car camping. Too much work, and at this point, the smoke makes it hard for me to breathe.
I did burn some of my trash in fires in the early days. Most of us did. When I hiked the JMT I would burn the pages of the book I was reading in a small campfire. The worst was a guy who decided to burn his sneakers in the fire - burning rubber really stinks.