In the context of camping, it's perfectly safe to wear a down jacket. Keep in mind that fleece is typically also made from synthetics, and so can be expected to have similar properties to your down jacket. (Actually somewhat worse, given the texture.)
A table of synthetic fiber characteristics at http://www.tensiontech.com/tools_guides/burning_characteristics.html provides the following information:
Both nylon and polyester will melt and burn in a flame, but stop burning when removed from the flame. Polypropylene and polyethylene continue to burn when removed from the flame.
For activities where there is a greater risk of exposure to heat or flame (e.g. hot air ballooning or glassblowing) it's common to require only natural fibers* such as wool or cotton. This is because synthetics melt as they burn, and can fuse to your skin which increases the severity of the burn. (Cotton actually has greater flammability than nylon, but has the benefit of not melting.)
Finally, a fabric's characteristics will also affect how it burns. A looser weave allows greater airflow compared to a thick, dense weave, and a fuzzier surface texture can make it easier for flames to propagate. (Example: using dryer lint for tinder vs. a piece of smooth paper.) The fabric may also have been chemically treated with flame retardant or DWR coatings.
*Nomex is also useful, but unless someone is a firefighter or race car driver, they are unlikely to have such a suit lying around.