Actually, as far as I can tell at a bit of digging (mainly Wikipedia, which confirms what I think I remember having previously learned), the two have nothing to do with one another.
A "Pyrrhic victory" is apparently named after Pyrrhus of Epirus, who won a battle against the Romans, but with such heavy losses that he commented (paraphrasing) "Another win like that and we'll have lost.". His name doesn't seem to be derived from flame in any way - or if it is, the fact appears to be purely coincidental.
Which doesn't mean that the author necessarily didn't intend a parallelism between the references in this case, however.
A "Pyrrhic victory" is apparently named after Pyrrhus of Epirus, who won a battle against the Romans, but with such heavy losses that he commented (paraphrasing) "Another win like that and we'll have lost.". His name doesn't seem to be derived from flame in any way - or if it is, the fact appears to be purely coincidental.
Which doesn't mean that the author necessarily didn't intend a parallelism between the references in this case, however.