While it's more common to refinish wood dressers with a clear finish than it is to paint them, there are exceptions. If the dresser has a damaged veneer, painting it is one way to hide the damage, and it's an alternative to stripping and refinishing a dresser that has already been painted. Moreover, some antique dressers are made to be lacquered with an opaque finish. Thankfully, preparation for painting does not usually involve stripping.
Choosing the Paint
Painting a dresser is more like painting interior trim than it is like painting walls. You'll need gloss or semigloss enamel, not regular wall paint. Whether you buy the paint in spray cans or in regular cans so you can brush it on is a matter of preference in most cases, but you should choose spray cans if you are repainting antique furniture with a smooth, previously painted surface. Be sure to buy a can of wood primer if the dresser hasn't already been painted. The primer will improve the adhesion of the topcoat.
Preparing the Workspace
Find a place where the dresser will be safe from curious hands and set it on top of a plastic sheet. It will be easier to paint if you remove all the hardware and if you have lots of room to spread the drawers out. If you plan to spray the paint, protect the walls with newspaper or plastic and be sure the room is well ventilated. The climate of the workspace is important. If you paint in a moist basement, you may have problems with slow drying. Spraying in a hot attic may produce a crackly finish.
Preparing the Dresser
After you've scraped off any flaking or peeling paint with a paint scraper or a rotary tool with an abrasive brush attachment, you'll probably find gouges and holes, and you can fill these with wood putty. If the dresser has a glossy finish, dull it with a palm sander and 120-grit sandpaper or an oscillating tool with a fine-grit sanding pad. The idea isn't to remove all of the old paint or finish, but to cut down the gloss and feather the edges left by paint or finish that has peeled off. This is also the time to make any needed repairs, such as gluing drawers back together or gluing down veneer.
Painting the Dresser
Brush or spray a coat of primer on new wood or on a finish that has substantially peeled off; then give it a light sanding with 220-grit when it dries. If you choose to spray your dresser, hold the spray can about 6 inches away and spray in long, straight strokes. Use the same long, even strokes if you choose to brush it, being careful not to lay the paint so thick that it drips. If you do get drips, let them dry before you sand them down. You'll probably need two or three coats. Sand in between coats with a minimum of 220-grit.