Rejecting the so-called "appeal to authority" is a tactic used by all kinds of "skeptics" to "debunk" ideas that they do not like. Most frequently I encounter this tactic in discussing God with people who claim not to believe in God. Usually, it turns out that I do not believe in the God they do not believe in either, because they are not actually discussing the Person whom I know as God, but that is another point entirely.
When someone appeals to authority, they usually put it in terms such as "Because [Person/Institution/Source X] supports [Truth Claim Y], then I support [Truth Claim Y]". Usually, in debates about God, it goes something like this "Because the Bible says [Y], I believe [Y]. God says it, I believe it, and that settles it." Nontheists rightly argue that this proves nothing, because the reliability of the Bible is still in question. It may say that Y is true, but how do we know it is accurate in what it says? I mean, there is the issue of who wrote the Bible and when, and whether these writings are authentic and accurate. Then there are the textual issues of how well the text has been preserved, even if the original text was accurate. And then there are issues of interpretation, linguistics, and historical-cultural context, not to mention genre and purpose, in considering how to understand the text. Then there is the question of what presuppositions we bring to the text, and if there are other legitimate ways of understanding the text. All of these must be considered before making an appeal to Scripture to support a certain truth claim.