Nearly any plant materials (especially seeds, fruits, and blossoms) can be ground into a paste useful for oil or essence extraction. The extraction processes are widely varied, but the part that varies the most is the difficulty of the extraction process and the yield from the plants.
Edit: I missed the post higher up about peat before I submitted this post. I still want to show my support for it as a fuel.
Not sure if this is feasible, but perhaps df could also have petroleum deposits or petroleum sources as a resource. This could vary from the different crude oil varieties all the way to oil shale layers. Perhaps methane clouds could form around oil deposits. If there were some way to drill for oil and collect methane gas, that would be interesting. Not sure about medieval fantasy mechanisms for harvesting, storage, or transport of gases, but it's an idea. Also, methane could be collected from livestock areas.
And one last tangent... The fuel/air mixture is a huge factor in the temperature of combustion. Many mechanisms for burning liquid fuel tend require the fuel to be aspirated prior to combustion. Interesting trick, a lit match dropped in a glass of diesel fuel will simply go out.
Someone mentioned a thermal lance higher in the thread. I've seen some thermal lances that used a mixture of diesel fuel and water as a fuel and shot out a flame about 2-4 feet out depending on the pressure of the fuel. Burning with water might seem counterintuitive, but it's actually necessary to keep the nozzle temperature low enough to be able to hold the shaft of the lance further down without burning through work gloves.