That is just ignorant. Really, really stupid.
1. The vines are already cut, and will not regrow. At all.
2. Kerosene won't do anything but preserve the vine from drying out. It will also increase the probability of spreading the urushiol by any casual contact with the vines. Urushiol is an oil, and kerosene will dissolve the oil in the plant and act as a carrier of the antigenic oil to anything that rubs against the vine.
3. As has been mentioned previously, burning the vines is hazardous to the health of someone allergic. Soaking in kerosene won't do anything to mitigate that risk. If you can wait for the log to dry out enough for the bark to fall off, it should be easy after a year or two to use a tool to pry off the bark and then throw it into a compost pile, or even burn it. Make SURE you throw the vine covered bark on a healthy fire, and then leave before you get exposed to any smoke from the vines. Upwind with gloves, hold your breath, and you'll be fine.
4. The vines are attached very firmly to the bark of a tree, especially the big vines at the base of the tree. If you are highly allergic, attempting to pull the vines off with pliers of some sort would be futile and very risky.
5. No one has mentioned being watchful for the seeds germinating. Anywhere that mature seeds have dropped, you should consider treating with a pre-emergent herbicide, or remaining watchful in the future and spraying emerging vines with a brush killer.
I have a suggestion for you highly allergic guys. Purchase a homeopathic product called Rhus-tox. These are little more than sugar pills with a minute quantity of the antigen that causes the rash. By swallowing very low doses on a daily basis, you can become immune to the poison ivy (or other plants that contain urusiol, including mangoes). It is available in a variety of concentrations. I recommend starting out with the weak stuff, and getting stronger with time.
The advertising for the Rhus-tox won't mention it as a treatment for poison ivy allergy abatement. That would be a medical treatment, for which it is not FDA approved. But it does work that way for a lot of people. Over the years, I have gotten quite a few of my guys so that they have only minor or no reaction to working in the vines.
Myself, I've always been immune. I did string trim a giant vine once while wearing shorts. My bare legs were completely green with the juices off the vine. I worked a whole day, and washed it off at the end of the day. I wasn't worried, 'cause I'm immune, right? I never got a rash, but legs itched like hell for a week.