As mentioned, wood will be totally fine when used under polymer clay ("covered" with clay), or just used with polymer clay, during baking as long as it's totally dry. Wood objects/shapes/etc are covered with polymer clay a lot.
However, since wood and other wood products like paper, cardboard, twigs, gourds, etc, are porous**, you'll usually need some way to give the surface tooth so the clay will stick to it while raw. That can be done with a previous coating of permanent white glue (PVA), acrylic paint, water-based clear finish, liquid polymer clay, etc. If the clay has a "mechanical hold" on the wood though (surrounds some part of the wood so it can just hold on) and it's able to stick well enough to put in the oven, that wouldn't be necessary but still is usually done.
Materials like the permanent white glue are good too because they provide a buffer between the clay and surface to reduce sudden temp rises and also are kind of stretchy themselves which reduce cracking in the clay, etc.
Those wood/paper products will not burn until they're exposed to much higher heat than we use, which could include being too close to the hot coils in an oven (much hotter than the air in the oven cavity). In fact, plain white or brown paper is used quite often as a baking surface for polymer clay (on top of something stiff to transport it into and out of the oven), and will only begin to brown a bit after many bakings. Wood blocks are sometimes used also to elevate baking trays so the clay can be as far away from coils, walls, and sides as possible.
There's loads more info on covering wood (and other wood products) on this page at my polymer clay "encyclopedia" site:http://glassattic.com/polymer/covering.htm
...look under the main categories called Wood
and Papier Mache, Etc.
and this page has more on paper used in certain ways:http://glassattic.com/polymer/mixing_media.htm
...look under Paper
** Cotton-based fabrics (and rayon?) are also porous, and there are lots of fun things to do with all kinds of fabrics + polymer clay and/or liquid polymer clay (see the Mixing Media
page just above, or the Sculpting-Bodies
page under Clothing
or the Miscellaneous page under Sewing
, if you're interested in any of those things).
P.S. Love your screen name
...look for a private message from me about some of the things you mentioned in your profile too