Remove the drawers from the dresser. If the dresser has a hutch with doors on the top, unscrew the hinges and remove the doors from the remainder of the piece. Disassemble the rest of the dresser as much as possible.
Sand all of the laminate surfaces with very fine grit sandpaper. Remove all of the shiny varnish from the entire dresser but do not sand through the laminate itself. Wipe down all of the sanded surfaces thoroughly with a rag to remove dust and debris.
Apply one moderate coat of all-surface primer to each laminate surface you will be painting. Stroke the primer on in straight lines that overlap as little as possible. Allow the primer to cure at room temperature for five to seven days.
Paint all of the laminate pieces with your chosen interior paint color. Roll on one thin layer of paint using a low-nap paint roller and allow it to dry 24 hours. Add a second, even coat of paint and let it dry another full day.
Cover each laminate piece evenly with spray-on polyurethane. Let the polyurethane dry 12 to 24 hours. Reassemble the dresser at the spot where you'll be using it.