Depending on where I am, the carcasses get dragged out of the center of fields to the edges, out of the way of the farmer's activities, or get moved to a designated disposal area.
As for gut piles, I don't know why anybody worries out them. Generally, they don't last more than a few hours and they don't involve anything particular tough or solid to foul the farmer's gear like bone or hide.
Same here, if it is in the way of Ag activity or other traffic, I will move it to a safer spot. Other than that stays where it lays. I used to hang coyotes on fences years ago, used to be a town talking peace, but as time went on, seemed to get more critical action (fences by the roads), now I just toss them.