That's critically important from a spiritual/philosophical point of view. The Eternal Flame can't be just any random bunch of plasma, it has to be That Particular Plasma.
I'm not being the least bit sarcastic. I respect and agree with the principle.
That said, my own interpretation would also include continuity of the fuel, since the flame's constituent components include material from the fuel. So transferring the plasma to a flaming stick, and then "something else", and eventually to another new Eternal Flame device with the appropriate fuel, has to be done properly.
Herd they declined to pay for fire insurance too!
Turns out that's not a quite accurate description. GoogleAI, quoting from the New York Times, says:
Because the monks manage the sacred embers separately and reacted quickly, the Eternal Flame itself was rescued entirely unharmed and temporarily relocated.This also implies that the fuel I mentioned in my comment above is some sort of wood, which of course makes sense in that context.
Aside: I find the concept of an "unharmed flame" rather charming.