It should be noted that nearly all scholars from the nineteenth century are outdated in some major way. In fact, even scholarship from as late as the 70s need serious reworking due to new data as well as new understandings. Thus the march of progress.
Despite this, there is some good in Frazer. Beyond being entertainment and a wealth of literary inspiration, the basic school in which it is centered, the Cambridge ritualist school, still produces a viable interpretation of ancient myth. Problem is, history, archaeology, and comparative methods have moved far beyond what Frazer was even capable of offering. Barbara Kowalzig's Singing for the Gods is a great new (well, 2007) book showing how to do it right these days.
But as for Frazer's book in particular, he is perhaps most outdated by his excessive speculation and assumptions. It is one thing to formulate a hypothesis about an origin of a particular myth or folkloric tale, quite another to ignore many differences between cobbled 'parallels' and make what are quite fanciful reconstructions of "savage humanity" from them, generalizing without controlling for locally-produced or contextually-driven origins. It's not sound scholarship to assume your version first and then assume divergent versions diverge from your assumed version. Frazer is guilty of this to the extreme.
On top of that, he adds details where necessary to fit his theory. Even the central thesis, as many over the years have noticed, is quite an imaginative interpretation of a very innocent events. The question he originally attempted was good enough, essentially "What's up with the priesthood of Diana of Nemi?" What he ended up with—priest-kings, scapegoats, and complex, magical rituals—is missing from the scene, which, of course, must be why he generally dropped it.
All this and we're ignoring the major shift in anthropology which realized long ago that the "progress from savagery to civilization" is a false progress (the "stages" are a great example of bad generalization based on one flawed sample), that primordial mythology of "savage beings" is imaginary, a tall tale of the Victorian world.
As for entertainment, it undoubtedly is, and its influence is legendary, but no, do not read it for an understanding the complex history of quite nearly anything he says. He has flashes of insight, but you'd be better off looking elsewhere.
If you want to look at some criticism of Frazer, I'd recommend these articles:
Beard, M. 1978. "Judgments on James Frazer." Daedalus 107.4: 151–164.
-- 1992. "Frazer, Leach, and Virgil Comparative Studies in Society and History 34.2: 203–224.
Leach, E. 1961. "Golden Bough or Gilded Twig?" Daedalus 90: 371–399.
-- 1965. "On the 'Founding Fathers.' " Encounter 25: 24–36.
-- 1985. "Reflections on a Visit to Nemi" Anthropology Today 2–3.
Smith, J. Z. 1973. "When the Bough Breaks." History of Religions 12.4: 342–371 (also found in his book, Map Is Not Territory).