Disposable diapers will burn. The inner layer and the outer layer are plastic which melts into a sticky mess that is difficult to remove from skin until it cools. Under certain circumstances, you can see burning drops of melted plastic which are extremely painful on bare skin. The absorbent material is mostly wood pulp which has been processed for maximum fluff and absorbency. The polymer will also burn when dry. If the diaper was used, the fire must evaporate the water before consuming the diaper. Yes, the polymer will dry out and burn with a good enough fire.
Since diapers will burn, some may be concerned that they are wearing a flammable garment. Don't worry, they are less susceptible to igniting than cotton garments (normal underwear). A standard lighter flame must be held to the diaper for several seconds before ignition will occur. Once ignited, the thick absorbent pad will insulate the genitalia from harm long enough for a conscious and unimpaired person to extinguish the flame. Areas only covered by the plastic or woven cloth-like layer will end up with plastic melted to the skin with second degree burns likely. If this does happen, try to avoid spreading the melted plastic and cool it quickly such as dousing with water. Smothering will press the plastic harder into the skin and help it retain heat. Prevention is the best method to avoid injury.
The environmental impact is a valid question. The fumes are harmful and foul. Normal disposal into a proper trash receptacle should be the primary means of disposal. Only in rare circumstances would burning become a reasonable option. Only during primitive camping situations far removed from proper trash bins have I seen diapers burned instead of thrown away. Some parents consider a weeks worth of diapers too burdensome to carry out and will resort to burning. With a proper fire, the entire diaper will be consumed and no traces will be left.
PS: The sources of information used for this post are old and not important. In other words, don't ask me how I know.