If heating oil is still a thing where you are, compare its prices to diesel (especially historically). They’re nearly the same product, the difference is almost entirely fuel taxes.
If you’re caught running red diesel on a normal carriageway, the penalty can be steep.
And it's not the same liquid. From first principles maybe, but not after it has gone through the supply chain until it reaches you.
Anecdotical evidence from decades ago, when I had an oil-fired oven for heating. With a big barrel in the cellar, which you let fill by hose from a truck maybe once a year.
And hand pumped into a can every few days, and carried that up, to fill the tank of the oven. One year I was rather broke, and that barrel empty. And then came winter.
Thought I'd be smart, because it's just the 'same stuff', right?
Walked to the Aral maybe 200 meters away, filled a canister.
That 'same stuff' made the fire drum clog from the inside with the finest carbon, like those mushrooms which grow like disks from the side of trees.
That never happened with just oil. And that was normal Aral diesel, nothing special.
One can also notice a difference in smell with older cars. Just oil stinks more.
Filter your diesel through cat litter in your garage and fill up your £60k mercedes.
Feels like extra work I would really enjoy doing and also the wonderful smell.
And I can probably re-use the cat litter for pesticides, so i don't need to worry about disposing of it.
But otherwise i would probably recommend against it.