(On a side note, Daniel Fincke is a badass, and anyone who is not familiar with Camels With Hammers should give it a look.)
(On another side note, if someone believed and took seriously certain statements attributed to Jesus, the most direct logical expression of that belief in Jesus would be to reject christianity (at least any recognizable form of christianity). That's not exactly the question here, but I think it ties closely into this question).
"Does faith itself lead to atheism?"
It can (it sort of did with me). But not directly. And not reliably. In a vacuum, belief leads to belief. People believe and then they keep believing. It's a simple enough equation.
But to believe and be disapointed in that belief, to have your belief confounded, to find out things that could not be compatible with what you believe, to have to examine your belief to find out if it is true, to find out it is not true? Yes some potential chains of events could take someone to disbelief as a natural extension of belief.
But what Fincke is describing here is not exactly "faith leading to atheism". More like "attachment to truth leading to apostasy". He's relating it to a religious obligation to search for truth. The main point seems to be that disbelief is not simply the result of a rejection of the god question, but the logical end to the search for god.
The thing is, at its broadest definition religion is a search for Truth. Not the search for truth. Philosophy is the search for truth. Religion and science are two distinct branches of philosophy which propose very different methods for defining and finding truth. Religions stress revelation as the source of truth (this emphasis on revelation is the one common theme of all religions). Early forerunners of the scientific method were explicitly a response to correct a flaw within the revelation based method of seeking truth (ie without testing, anyone can just assert whatever the f*ck they like as revelation, and there's no way to tell).
Ask anyone and they will say they value the truth. Few will be consciously lying about that part. The question is one of how we define truth, where do we look for it, how do we distinguish true from untrue, and by what reasoning we declare something to be true.
As FO mentions faith =/= love for truth. Love for truth (or at least love for "truth") is part of the religious mindset. But not the same part as faith (though this usually won't come up if the person is not consciously aware of any reasons their faith might be misplaced). Faith is committed attachment to the declared truth value of a proposition. It is by nature indifferent to the value of truth.
Faith is often described as belief without evidence. It is that, but the 'without evidence' part is a byproduct of what faith really is. Faith is unconditional belief. With or without evidence. Indifferent to evidence. To really have faith is to not care at all whether the evidence is on your side. What is important is not what is true. What is important is that this is true.
If you have faith not only that your premise is true, but that it is the pinnacle of truth, and the prime basis by which truth is measured, it is very hard to ever seriously consider that it may not be true. Why would you? If some other thing seems to contradict that premise, then that other thing must not be true. How could it be true if it disagrees with the bible/Qu'ran/Book of Mormon/etc?
Most major religions specifically teach that anything from outside their sphere is not to be trusted. There are many barriers to exiting an insular position. But if your starting point is an obviously wrong dogma, and searching for truth is something you take seriously, it should lead to apostasy (and in this equation "apostasy" does not necessarily mean "atheism").
Where faith itself can lead (indirectly) to atheism:
Within the portion of the populace which would be classified as believers, there are those who believe, and there are those who are just there. Those who are just there are just there by default, and unless they get bored with church, they'll basically always be there.
If a person believes in god because why not it's something to do on Sunday, and doesn't really take this god thing seriously or care whether it's real, they can potentially believe forever without being challenged in their belief. If a religious person finds religious ideas too boring or confusing to examine they will encounter fewer flaws in those ideas. Anything they happen to encounter which does challenge their beliefs can be easily shrugged off without even the need for rationalization, since belief is more of a social thing than a belief thing anyway. Because they lack faith, they will continue to have faith.
If a believer really believes (which is to say, if they have faith), they may take spiritual questions more seriously. If a person takes spiritual questions seriously, they might be more concerned with actually asking the questions and trying to find the answers. They may be more driven to actually examine their faith. Most of these will become zealots or apostates. Those who examine the claims of christianity, and the basis of those claims, and the history of the church will find many reasons not to believe. Those who get this far will come to a fork in the road. A choice (which may not even be consciously perceived) must be made. Either it is more important to believe what is true, or it is more important that what you already believe is true. If what you already believe in is following the truth wherever it leads (which some would consider a religious tenet), the choice is that much easier.
I've often said the best reasons (and my main reasons) for rejecting christianity come from within christianity (and its associated literature). There is no shortage of outside evidence. But when faith is contradicted by outside evidence, it is still valid according to its own standard. When faith fails on its own terms, that is when faith goes away. I started as a believer, and I followed that belief to its logical fulfillment. I looked for god, and no god was what I found. Faith can lead you to that. If you don't have too much of it.