I'm thinking emissions with veggie products , especially filtered waste veggie, would always trump burning fossil varieties because it's carbon neutral. ie. the restaurant is going to use veggie no matter whether I collect it and burn it for fuel or not and thereby I burn less fossil products. An additional thing to consider is that the vegetables---let's say corn--have been consuming CO2 while it grew and the plants that were planted to replace those plants continue to consume CO2 as part of plant photosynthesis. Fossil fuels are just that, fossils so they never use CO2 for anything as they are long dead. So emissions from burning plant oils might trump others sources from an environmental perspective for these reasons.
Another factor I remember reading somewhere is that diesel combustion particulate is large compared to petrol and falls back to the ground instead of the petrol fumes that vaporize.
I'm not an expert but this my understanding. I'm interested in other perspectives and learning more about it.