There's a grand total of two threads on this subject on this sub, both old. The picture they present is that flaming arrows have only occasionally been useful:
/u/Freevoulous argues that incendiary missiles are a historical rarity precisely because they're likely to be extinguished, but cites a couple of 14th-15th century examples
/u/GBFel cites some isolated examples of incendiary missiles in antiquity (note that the words for missiles used in the ancient sources do not specify arrows; firepots are at least as likely)
The second link cites a rare example of an actual published work on the subject, Adrienne Mayor's Greek fire, poison arrows, and scorpion bombs (2003). The relevant section is chapter 7, where Mayor shows a handful of examples that approximate to 'flaming arrows' or which she suggests imply their use. However: (1) The chapter is strikingly sparing with references to specific passages in actual ancient documents; for example, when she claims the Persians attacking the acropolis of Athens in 480 BCE 'shot fiery arrows', she doesn't mention Herodotos, the source for this claim; she doesn't cite the specific passage (Herodotos 8.52); and she doesn't specify how they contrived the flaming arrows, even though Herodotos does. (2) I spotted occasional false claims on minor points. (3) Nearly all of the examples she cites aren't arrows, but things like handheld burning torches, fires ignited next to a city wall, combinations of pitch and burning kindling, and other contrivances. The book isn't dishonest, but it isn't thorough and it isn't precise.
Mayor does cite Alfred Crosby's Throwing fire (2002), but that's even more lacking. For what it's worth, the relevant section in Crosby's book is chapter 5 'From weapon craftsmanship to weapon technology', which sets out to cover everything between the early Palaeolithic and the gunpowder age in 23 pages, and cites barely any ancient/mediaeval sources.
In short, there's little scholarship to cite on this subject. That probably speaks to the ineffectiveness of flaming arrows.