Burning used transformer oil as an on or off-road fuel is illegal. If one gets caught taking transformer oil off site and using it as a fuel, well, I'd rather get caught using off-road diesel. That would be a slap on the wrist compared to getting caught using used transformer oil. Now I'll go into some of the details.
I've worked in the electrical generation business for over 30 years now. I work around transformers that are the size of a small dorm refrigerator to the size of a two story house. The really large ones hold over 5000 gallons of oil and have banks of recirculation pumps and coolers.
The new transformers use PCB free oil. Older transformers are flushed and the new oil is PCB free. BUT, there are thousands and thousands of old transformers out there that still have PCBs in them. The EPA made using PCB treated oils illegal in 1979, but that does not mean that overnight all the utilities went out and flushed or replaced its transformers. Replacing small transformers and flushing large ones is done on an as needed basis so there are still a lot of transformers out there with PCB treated oil.
The BASE oil would make a great fuel. The base oil is a highly refined hydrotreated very pure mineral oil that is high in naphthalenes. It would make an excellent solvent. However, there are still additives in that oil that I would be weary about burning in any engine. There are several different additives used for flame retardant, thermo-stability, and suppress aching and corona. Arching is the electrons jumping from one place to another, usually in air. Corona is when the surrounding gas around the arch is superheated and turns into plasma. This can also be a liquid (like an insulating oil) that boils into a vapor, and then that vapor gets superheated into a plasma. Some of these additives contain silicates, something you do not want in your engine.
The myth is that PCB free oil is safe to burn. The truth is that the additives used in place of PCBs are not safe and can cause any health problems PCBs can. An analogy is asbestos Vs rock-wool (a type of fiberglass). One is banned now and the other is not. Rock-wool will destroy your lungs as quick and easy as asbestos can.
Even though a transformer is a passive device and has no moving parts, they do age. The insulation between the coils slowly breaks down. Motoring is the usual stuff like thermocouples to measure temperature at several points. N2 gas blankets the transformer to prevent any air leakage.
Several of the large transformers use gas monitoring. If there is any arching inside the oil is designed to give off acetylene and other gases. These are tell-tale gases in which samples are taken and analyzed. If a transformer has excessive 'gassing' then an outage is scheduled to fix problems before there is catastrophic failure.
Used transformer oil would contain a large amount of contaminates. There is a very good reason the utility spent a lot of money to remove it. Chances are it will also be contaminated with PCBs. Most utilities store used PCB free oil in the same containers they use for oil with PCBs. Utilities dispose of transformer oil by burning it in special licensed incinerators. The main criteria being the incinerator is away from any population and it has to burn the oil at a temperature at least 2000F for a certain time period. Incinerator burners are simple low pressure atomizers and they give the oil plenty of time to burn off completely. A diesel engine does not.
Sooner or later someone is going to post they have done this and their engine runs great. The high naphthalene content will burn clean and the user might notice less soot, however, that does not mean there are not other nasty things coming out the exhaust pipe. The exhaust might have less soot, but it will have many other contaminates not found in diesel exhaust.
If one wants to know more, just type 'burning used transformer oil' in your favorite search engine. You will have plenty of reading to do.
As an aside, if you have utility poles with transformers mounted up there, those are 'wet' transformers that do contain oil. If you live in a more modern subdivision with underground electric service and you have those green transformer boxes humming at the end of the block, those are 'dry' transformers. There is no oil inside those.
Only plant oils are used for biodiesel and SVO/WVO.
I hope this dispels some of the misinformation.