Look for screws in the end units. Not all porcelain towel racks are antiques. Newer racks may have screws in the end units, which are on either side of the towel bar.
Use a screwdriver to carefully remove the end units. Be sure to catch the unit and the bar once the screws are removed, as both of these things may slip suddenly from the wall. It might be a good idea to line the floor with towels so that if the bar does land on the floor, it won't break.
Slide the end units up and down. If there are no visible screws in the end units, this means the units are "hooked" on screws through holes in the back. Sliding the units up and down should release them from the screws holding them in place.
Pry the end units off. If they will not slide, and you don't see any visible hardware to remove, use a thin, flat screwdriver to slip under the end units and try to pry them off. If the towel rack is very old, some paint may also come off the wall, so be prepared to do some touch-up painting.
Remove all parts of the towel rack once the end units have been loosened--which in most cases will consist of two end units and one towel bar, along with any hardware used to hang it.