If two people start with totally different fundamental assumptions, one will never convince the other. I mean truly fundamental assumptions, such as assumptions about what even counts as evidence. Because of this, the most extreme religious dogmatist, no matter how stubborn, may not suffer any psychiatric illness or make any egregious error of deductive reasoning.
Note that the situation of such people relative to others is not symmetrical. In our everyday lives, we have standards of evidence that most people - religious or otherwise - accept. E.g. the fact that I can see no hippopotamus in my study when I look around is good evidence that there is no such hippopotamus in my study. If my lovely Significant Other pops in here in a minute, I can ask her if she sees one, but she'll think I've lost my mind (or, more likely, that I'm asking her as part of a philosophical thought experiment). Outside of religion, we have no trouble reaching consensus on most everyday issues because we use common standards of evidence and possess similar powers of observation.
Science typically deals with things that can't be observed directly with our unaugmented senses - often because these things are very small or very far away or no longer in existence. We have traces and indirect clues, but no one has ever observed, say, a brachiosaurus, only certain fossilised bones (and even the interpretation of them as fossilised bones requires a body of theory). But science uses methods that are continuous with our usual methods for inquiring rationally into the world around us. Over the past four hundred years, science has been able to tell us much about the world beyond observation by our unaugmented senses. It delivers reliable knowledge by using such methods as scientific instruments, highly precise modelling, conducting experiments that control for extraneous variables, using convergence of results from many lines of inquiry. Most of these are methods that we use in everyday life, but outside of science we don't always need to be so rigorous because everyday questions can often be resolved just by observing.
Applying ordinary kinds of reasoning from everyday life and from scientific inquiry, we can answer many questions. People who disagree can be brought to change their minds if the evidence is provided to them. Over time, science tends to converge on answers and to develop well-corroborated theoretical knowledge that is unlikely ever to be overturned, such as the knowledge that life on Earth evolved over hundreds of millions of years. That doesn't mean that the scientific image of the world is ever complete; there's always new stuff to find out. Still, there are many things that we can now be confident about - the approximate age of the Earth, the evolutionary explanation for the diversity of life and the apparent purposiveness of adaptations such as the eyes, the descent of Homo sapiens from earlier primates, and so on.
The people who can't be argued out of their position are those who have certain substantive beliefs about the world that go psychologically deeper than their commitment to ordinary standards of evidence. This certainly includes some religious people - the so-called true faith-heads. If someone believes doctrine X in that way, and is then confronted with good evidence (by ordinary standards) for ~X, she has many options including the option of claiming that ordinary standards of evidence are wrong or inapplicable. Someone like that can enjoy a hermetically-sealed worldview. No argument based on evidence can ever penetrate it.
But notice that this person need never be irrational in the sense of refusing to apply the basic rules of deductive logic, or in the sense of refusing to apply ordinary standards of evidence in her everyday non-religious life. Nor need she be suffering anything like a psychiatric illness. Still, there's a sense in which irrationality is present.
Most atheists are open to evidence that God exists if ordinary evidence is supplied. In fact, there'd probably be no atheists around in Western countries if diluvian geology had turned out to be correct - i.e., if dating methods pointed to an age of the Earth of 6000 years, and if the fossil record and the facts about rock formation turned out to be consistent with a mass extinction in Noah's flood and the formation of the rocks and the fossil record at the time. Even now, there is probably evidence that could (in principle) come in to persuade an atheist to change her mind, though the historical record of the religionists is so dreadful that it will now take something remarkably compelling.
True faith-heads, by contrast, are immune to evidence. Since they are committed to certain substantive theological claims more deeply than they are to ordinary standards of evidence, they will even develop revised evidentiary standards if that is the only way out. In between, they'll clutch at all sorts of propositions that seem unhinged by ordinary standards. E.g., we see YEC Christians committed to a 6000-year-old Earth who will admit that the Earth looks 4 to 5 billion years old (using various dating methods)... but of course, God had His reasons for making it look that way when He created it 6000 years ago. One such person is currently arguing the toss about this over on Richard Dawkins' site. I can't disprove what he is claiming, but when it's viewed from outside it's plainly unacceptable.
There's no arguing with such people. All you can do is point out how they think and why that kind of thinking should not be given any credibility or respect by anyone who isn't already infected with it. You can also point out how arbitrary the deep assumptions that these people make really are and how baroque their worldview eventually becomes when they are pressed. But, in the end, they either see how unsatisfactory the entire picture is or they don't. No single argument can penetrate their sealed-off view of the world.
How many religious folk are true faith-heads in this sense? Well, I suspect it's a kind of sliding scale. It seems to me that it would be very difficult for a sincere and inquiring person to hang on to some religious doctrines unless they go very deep in her set of beliefs. It may take a lot of true faith-headedness to stand up for Young Earth Creationism in the face of all the scientific evidence. But I suspect that a fair amount is required even for belief in a loving and providential (and all-powerful, all-knowing) God. There is always a way out, to seal off such beliefs from ordinary standards of evidence, but if those standards are accepted it is not believable that such a being would have allowed the millions of years (and enormous amounts of misery in the animal world) needed to evolve us, or would have made its own presence in the world so un-apparent, or would have intervened at such a late stage in human history.
Now, one can argue that God must have had a reason for these things. E.g., maybe God is so un-apparent because He values belief in his existence by those who believe despite the lack of evidence. Maybe those who believe in God are just those whom God has spoken to personally. Why not speak personally to everyone then? Well, maybe only some are receptive to the message (and God would know who they are). Why not make everyone receptive to the message, if you happen to be God? Well maybe you have a choice, if you happen to be human, whether or not you're receptive. Why not make all of us so that, given such a choice, we'll want to be receptive? Well, free will doesn't work like that...
By now we're moving toward a baroque and implausible idea of free will, what it means to choose to be receptive to a message from God Himself, and so on, but for the dogmatist, there's always room to move.
Some people are beyond arguments based on ordinary standards of evidence, and they cannot be reached. The most we can hope is that they will eventually work out that the entire edifice looks wildly implausible from outside, and that that's a good reason to reject it in toto, especially if they realise that the deep assumptions they have been making were shaped by culture and are really rather arbitrary. If they don't ever get to that point - one that involves seeing through the effect of their own socialisation - we can merely point out to others that there's no reason to give any special respect to the beliefs of these people. There's no reason to adopt those beliefs yourself unless you already share the assumptions of the faith-heads. The hermetic sealing keeps ideas in as well as out.
Again, the sort of irrationality involved need not include making a clear mistake of logic or being unable to navigate safely around the world. It needn't include hearing voices or anything of that sort. Nonetheless, it involves distorting your thinking by giving an unjustified place to certain arbitrary assumptions that other people would not (and should not) make on the basis of ordinary standards of evidence. Those assumptions are used as premises, in effect, rather than drawn as conclusions from everyday observation or well-corroborated scientific theory. This is not ordinary irrationality, and yet there's a point in calling it irrational.