Note: This is the beginning of an article written by Louis Tofari of Romanitas Press. Romanitas Press has published several fantastic items which I have recommended previously (see the reviews). I wanted to again recommend this publisher.
An easy way to remove the gummy incense resin build-up from a thurible is to soak it in carburetor cleaner, a degreasing solvent that can be obtained from any auto parts store. I recommend the 1-gallon can of carburetor cleaner that includes a submersible parts tray. The parts tray is handy for soaking the chains and other smaller parts (should you decide to take the thurible apart), while the height of the 1-gallon can allows for submerging most thuribles (should you prefer not to take it apart). For taller thuribles, a 5-gallon can is also available.
Carburetor cleaner is safe for thuribles made of solid brass, stainless steel, copper or silver. It is also safe for some plating finishes, such as nickel and silver - though I suggest testing an inconspicuous spot first on a gilded thurible (particularly of gold), before immersing the entire thurible or applying solvent on it. Mineral spirits and rubbing alcohol are also good degreasing alternatives.