There are a couple problems with this system:
- extra fuel is used to burn the particulates, which reduces overal MPG.
- extra CO2 emissions are released when burning up the particulates.
- Biodiesel, which vaporizes a higher temperature than petroleum diesel, doesn't function the same way in the system, and fuel ends up in the engine crankcase rather than clearing the DPF. 
 It depends on how the system is designed, of course. The problem is caused by using the fuel injectors in the cylinder to place additional fuel into the exhaust gases. Since biodiesel doesn't vaporize per the system design, some of it stays in the cylinder and leaks past the piston rings into the crankcase. If the additional fuel is squirted into the exhaust pipe, and not the cylinder, then you bypass the problem of fuel getting into the crankcase. But that is more complex and maintenance intensive (additional dedicated fuel injectors).