In brief, the problem with the FAQ is that it often answers these frequently-asked questions incorrectly. Even though it is an “official” publication of Wizards of the Coast, it is not, and was never intended to be, a source of authoritative rulings or carefully-considered implementation suggestions. It was intended to be a quick source of clarification, as any FAQ generally is, and answers were generated quickly – and so errors crept in.
Due to these errors, and the FAQ’s status as an FAQ rather than errata, its use in debates about the rules is generally regarded as pointless. It lacks both the authority and the credibility to serve as useful evidence for any position. That isn’t to say that good rulings cannot be found there – I agree with many of the rulings in the FAQ – but rather that a ruling in the FAQ has to stand on its own merits; it gains little-to-nothing for being “from the official FAQ.”
For more detail, I feel it is necessary to expand on just what the FAQ is and is not, what the rules themselves say about it, and thus develop an argument for why it is important to not give the FAQ undue weight.
It is perhaps worth noting that everything in this answer applies equally-well to the Rules of the Game articles found on Wizards’ website. Those articles are in a very similar category to the FAQ.
The purpose of the FAQ is to explain the rules – not write them
The purpose of the FAQ is to answer questions about the rules, based on those rules. It is not the place for fixing, caveating, updating, qualifying, or editorializing about the rules. Wizards of the Coast provided the FAQ as a hopefully-helpful source of expanded clarification on the rules.
This was not a high-value purpose for Wizards; mostly, it was a bit of community-outreach. The FAQ was written by one person, who had other, more significant responsibilities, and appears to have seen little-to-no oversight or editing.
Wizards of the Coast did have an official channel for updates – errata
On the other hand, official errata files were released for many books (though far from all, sadly). These officially update the rules, changing the wording here and there. These were released as downloadable .pdf files, and the tone and style alone conveys the difference between the errata and FAQ: the Sage muses about the rules in the FAQ, but the errata is all business, surgically editing out and adding in text.
As such, the FAQ was given no authority to change the rules under the rules
The errata documents themselves include an explanation of how they work, and how rules discrepancies between different D&D products were to be determined. Most relevantly here, the default rule for separate publications that contradict one another is that the “primary” source trumps all others – except its own errata.
For example, from the errata from Player’s Handbook:
When you find a disagreement between two D&D® rules
sources, unless an official errata file says otherwise, the
primary source is correct. [...]
Another example of primary vs. secondary sources
involves book and topic precedence. The Player’s
Handbook, for example, gives all the rules for playing
the game, for playing PC races, and for using base class
descriptions. If you find something on one of those
topics from the Dungeon Master’s Guide or the
Monster Manual that disagrees with the Player’s
Handbook, you should assume the Player’s Handbook is
the primary source.
Thus, errata documents were given the authority to change the rules.
The FAQ is not mentioned explicitly in the errata, but it is clearly not the primary source on anything – the books it is discussing are. As such, any contradiction between the FAQ and a published book is always going to officially be in the book’s favor.
Which makes sense, because the FAQ consisted of quick answers to questions, not carefully-considered design changes the way errata were (supposed to be). Instead of the book’s authors and editors reconsidering their work, the FAQ was one guy skimming the appropriate rules and making a quick judgment. Many FAQ entries weren’t even made by any of the book’s authors.
Finally, the FAQ itself was often inaccurate
As I’ve already established, in any contradiction between FAQ and a published book, the book was always primary and therefore “won.” But contradictions of this sort were very common – which means that even when the FAQ is right, you have to double-check it against the actual rules, to make sure it’s not a case where the FAQ was wrong. That makes the FAQ worthless for its stated purpose—since it cannot be relied upon and you have to double-check it against the published rules anyway, you might as well just start with the published rules and ignore the FAQ entirely. That saves you time and eliminates being misled by poorly-considered answers.
Ironically, the FAQ is at its best when it admits up-front that the official rules don’t cover a topic, and suggests a houserule. These avoid misleading readers, and occasionally provide useful ideas.
But in other cases, the FAQ gave no notice that its statements were in contradiction with the official rules – and indeed, the author was probably unaware of that – and then those statements, if applied in a game, cause problems.