Did some digging, got an answer.
Long story short, heat and chemistry turn the soot particles into carbon dioxide and carbon monoxide. What can't be burned off are the metals used as lubricants in engine oils, which create ash - hence why Ultra Low Ash oil is so important in DPF vehicles. This will typically just be blown out in very small amounts over time, but if not well managed, will cause the DPF to clog and create the white smoke associated.
The copy/paste of the more technical version:
The popular CRT device relies on the principle that particulate matter can be oxidized or "burned off" at lower temperatures in the presence of nitrogen dioxide than in oxygen. It uses two chambers. The first contains a substrate coated with the precious metal platinum. The platinum acts as a catalyst to convert nitrogen oxide in the exhaust stream to nitrogen dioxide by oxidation. The second chamber contains the ceramic filter. When the engine reaches operating temperature and the nitrogen dioxide from the first chamber passes through the ceramic filter, the collected particulate is "burned off". This "burn off" reduces the particle material to gases like carbon dioxide and carbon monoxide and various types of inorganic dusts ("ash") such as iron, zinc, copper and calcium oxides.
This was a pretty handy video too:
Forced DPF Regen question
