An interesting discovery, though tending somewhat toward the space-batty side.
A lot of the ancient writers on Greek Fire, mention tartar. Which seems odd, because tartar, potassium tartrate, is a very inert innocuous substance.
Then I discovered this note in a work from 1830
M. Brunner, by acting on calcined tartar in a. bottle of wrought iron, has succeeded in obtaining potassium at a comparatively moderate heat. The bottle is spheroidal, about half an inch in thickness, and capable of holding about a pint of tartar; a bent gun-barrel of ten or twelve inches in length screws into the mouth of the bottle. The bottle, well luted over with fire-clay, is set in a strong air furnace, so that the tube may dip down externally beneath the surface of naphtha contained in a cylindric copper vessel, sitting in a tub containing ice and water. The lop of the naphtha vessel has a cover fixed on it, pierced with a hole to receive the end of the (run-barrel; and, from the side of the upper part of the vessel, a small tube goes off at right angles to let the air and vapors escape. It is advantageous to mix a little ground charcoal with the tartar previously calcined in a covered vessel, in the same iron bottle for example. Nearly 300 grains of potassium have been procured by this apparatus from twenty-four ounces of crude tartar.
So. To make metallic potassium. Wrought iron. The Greeks had that. Gun barrel, no, but they certainly could have ad an iron tube, which is all it is. Fireclay, check. Naptha, yes. Charcoal, of course, and strong air furnace, yes.
So, the Greeks did have all that is required by M Brunner. To make potassium metal. Which spontaneously decomposes and burns with a hot red non-extinguishable flame when put into water. But, potassium and acetic acid would react to produce safe, inert potassium acetate.
I dunno. Awfully far fetched. Ancient Greeks making a metallic alkali ? Sounds ASB, but if the above quotation is true, then it would have technically been possible.
EDIT. Clearly, Greek Fire was NOT just potassium metal. It must have largely been some sort of hydrocarbon , or hydrocarbon distillate, and a resin of some sort. But, with the addition of some "special" ingredients, that gave it its particular fearsomeness. Use of burning oil , turpentine etc in sieges was common and well known. Nasty, but a commonplace nastiness. Greek Fire had to have something else, quite special, to give it its reputation.
And, given the way it disappeared quite suddenly, one suspects a "special" something that was rather difficult to make.