After a bit of research, most of the world seems to be on the same page when it comes to how different finishes can affect your sound. Of course your finish is only a small piece of the sound puzzle, it DOES affect the tone. This applies to drums, acoustic guitars, electric (in some cases) guitars, violins and other stringed instruments as well.
The general rule is this-the thicker and stiffer the finish, the more muffled the sounds produced. This is because the thicker/stiffer the finish is the more it prevents the wood from vibrating and echoing. Not every guitarist/violinist/drummer is looking for exactly the same sound; many prefer the more muffled sound and players must consider the durability of the finish vs. the sound quality.
Higher quality violins (as well as other stringed instruments) tend to be finished in a very thin shellac which produces a natural, woody sound. Guitars, especially the more user-friendly models (read: inexpensive), tend to be finished with either a lacquer or thin polyurethane which provides minimal dampening and more importantly, protects the wood from damage.
Drum kits can come in a variety of finishes as well,ranging from wraps to oil finishes to lacquers. A wrap is more or less a colored piece of plastic with one glue anchor point. Wraps tend to produce a more deadened thumpy sound, and are replaceable when damaged. Drums made out of more expensive woods such as maple, mahogany etc are often covered in stains or lacquers to allow the wood to resonate, but are much more fragile than their wrapped counterparts.
While finish is often a huge factor when choosing an instrument for aesthetic reasons, there are plenty of other things that can affect sound quality. Many many many many many other factors are at play here! String quality, nut, bridge, saddle, bindings, wood quality, wood type, age, humidity, how often it is played and finish-these things combined are what produce a unique sound!
Thoughts? Questions? Concerns/additions? All are welcome!