Cut through the caulk around the bottom and top trim on the inside of the coffered ceiling box. Slide a wide putty knife under the edge of the trim and pry it away from the box.
Unscrew the top of the box from the cleat with a screwdriver. Typically, the box that you see in the room is a faux beam. This three-sided box is screwed to a cleat along the top sides of the box. When you unscrew the sides, the entire box should be removable.
Place the wide putty knife flat against the ceiling alongside the ceiling cleat. Usually, the cleat will be nailed into the ceiling joist. Slide the claw of a hammer along the putty knife into the joint and try to loosen the nails. The putty knife will act as a shield and help protect the drywall ceiling from dents as you pry. Move the putty knife along the cleat with the hammer and pry bar as you loosen the entire cleat. Pull the cleat straight down to remove it. Repeat across the entire ceiling.