I can't provide a solid answer with citations on this - but it does look quite possible it was actually DeNiro doing the opening shot of this stunt.
On the insurance aspect, you might have to ask Tom Cruise how he gets away with being strapped to the outside of a plane, or in the upcoming Top Gun, actually flying one!
The shot only lasts two seconds so is not beyond the realms of possibility before modern 'health & safety' took over the world. The rest will be a stuntman in full protective gear, including a fire-proof full head-covering; which is why he has to be completely engulfed, so we can't see the 'rubber head', but that first two seconds would be quite possible to do using Zel-Jel, which was the industry-standard for fire protection at that time. I don't know if it still is, it's outside my area of expertise, but it was definitely used on Cape Fear.
Essentially, Zel-Jel is a two-part structure.
The first, a thin liquid which is pre-soaked into the clothing then allowed to dry.
The second component is a barrier cream applied to the skin, which not only will not burn but will also keep the skin cool during exposure.
The stunt fire is then set using a flammable gel applied to the clothing and lit remotely with a small electrical charge. The gel won't run or smoke and will have a quite specific burn height, so the fire area is very tightly controlled.
Juliette Lewis' lighter fuel is, of course, non-flammable. If you've ever seen someone getting that trick wrong whilst trying to get a barbecue going you'll understand why that element of random danger would not be a good idea.
You'll notice the fire area doesn't actually start from the cigar and lighter, it starts out towards his shoulder then spreads, whilst never quite reaching his face. As soon as the fire starts, he turns away from camera into full profile, making the flames look much closer to his face than they really are. They're not a mile away, but just slightly further than it probably looks. The angle of the shot and the quick cuts give us no real chance to spot this other than frame by frame.
By the time we cut away then back, we never see his face again, so that's where the swap to a fully protected stunt man can occur.
A late thought with no concrete evidence, however, it's quite possible that his hair design in this scene helped the stunt. As it's 'slicked back' presumably as the plot has him crossing the water to reach the boat, so his hair is wet, it also means they can do his hair in the Zel-Jel, thus avoiding having to wig him for the scene.
The fire crew will have been barely out of shot the entire time, and he'd be extinguished by about 4 seconds. They really don't get far from the action on this type of stunt.
The fire crew have authority over everyone on set, from the director down. No-one out-ranks them.
These days they have 'point & shoot' temperature sensors constantly pointing at hot areas and will pull up the entire set if they feel anything is going outside accepted parameters.
I saw this happen a few times when we were shooting the fire scene at the Medici circus in Dumbo, which took two weeks to get. The fires in that scene are real.
I may have to partially retract the 'playing fast and loose with real lighter fluid' comment from before. Whether they did or didn't use real fluid with DeNiro [which would honestly be utter madness with an actor's face right there] once the 'rubber head' goes on the stuntman, it seems they're no longer worried. I just caught this one frame of back-flare .. after which she rather sensibly throws the can to the floor
Re: comments on the type of accelerant
This, from Canadian fire stunt team Fire for Hire
What type of accelerant do you find works best with this gel?
We can use anything you want. We’ve used rubber cement, gas, diesel and we keep testing it every week with new fuels. We also experiment with the thickeners that we add to the fuel to make it burn longer. There are all kinds of tricks that we use in preparing the fuel and applying the gel to get the best burn possible.