Let's consider also the defintion of "irrational". From Free Online Dictionary we have:
a. Not endowed with reason.
b. Affected by loss of usual or normal mental clarity; incoherent, as from shock.
c. Marked by a lack of accord with reason or sound judgment: an irrational dislike.
a. Being a syllable in Greek and Latin prosody whose length does not fit the metric pattern.
b. Being a metric foot containing such a syllable.
3. Mathematics Of or relating to an irrational number.
An irrational number.
Clearly, we aren't talking about definitions 2 or 3. In the context cited above, 1c is the operative definition. If someone says to me that they believe in talking cats, and yet they can provide no evidence for it, they just continue to believe it anyway, that is an irrational belief: it is not in accord with reason. Similarly, if someone tells me that they believe in god, but they can provide no evidence for that belief, then I am equally justified in observing that their belief is irrational. If a believer makes an emotional argument, or because that's what they were taught, these are also irrational beliefs. One does not believe in the law of gravity because you "love" the idea. One does not believe in algebra because your teacher said it works. One believes in gravity and algebra because they work, and if I drop my pen on the table or try to solve an equation, and can provide more evidence that they work. A belief in god may be important to the person, but they have essentially already admitted that their belief is not based on evidence, and thus is not rational.
If you value feeling over reason, then I don't see why being accused of being "irrational" would be perceived as a bad thing. "Irrational" is only an insult if "rational" is perceived as a good thing. But rationality is a standard to be lived up to, not simply given lip-service.
To further clarify lest their be any attempt at equivocation, what I mean by "reason" is also found from the Free Online Dictionary:
1. The basis or motive for an action, decision, or conviction. See Usage Notes at because, why.
2. A declaration made to explain or justify action, decision, or conviction: inquired about her reason for leaving.
3. An underlying fact or cause that provides logical sense for a premise or occurrence: There is reason to believe that the accused did not commit this crime.
4. The capacity for logical, rational, and analytic thought; intelligence.
5. Good judgment; sound sense.
6. A normal mental state; sanity: He has lost his reason.
7. Logic A premise, usually the minor premise, of an argument.
v. rea·soned, rea·son·ing, rea·sons
1. To use the faculty of reason; think logically.
2. To talk or argue logically and persuasively.
3. Obsolete To engage in conversation or discussion.
1. To determine or conclude by logical thinking: reasoned out a solution to the problem.
2. To persuade or dissuade (someone) with reasons.
The definition which is relevant here is definition probably defintion 3. Again, I'm not referring to the person; I'm referring to the incoherence of their argument.
And seince we are talking about someone being logical, the definition of logic:
1. The study of the principles of reasoning, especially of the structure of propositions as distinguished from their content and of method and validity in deductive reasoning.
a. A system of reasoning: Aristotle's logic.
b. A mode of reasoning: By that logic, we should sell the company tomorrow.
c. The formal, guiding principles of a discipline, school, or science.
3. Valid reasoning: Your paper lacks the logic to prove your thesis.
4. The relationship between elements and between an element and the whole in a set of objects, individuals, principles, or events: There's a certain logic to the motion of rush-hour traffic.
5. Computer Science
a. The nonarithmetic operations performed by a computer, such as sorting, comparing, and matching, that involve yes-no decisions.
b. Computer circuitry.
c. Graphic representation of computer circuitry.
The relevant defintion here is defintion 3, or possibly 1. There are good reasons to believe something, reasons that are good regardless of whether one is arguing about god or a talking cat. And there are bad reasons, which people think they can get away with when talking about god, but are patently absurd when referring to talking cats.
So now, if I say that a believer comes to me and says that they believe in god because of an emotion or the like, that is irrational. (Never mind the irrational character of the gods they usually propose to exist.)
I am sometimes attacked, as is Dawkins, as was Carl Sagan, for believing that there is life in the universe besides ours. Do I have any direct evidence for it? No, and so I will say that I believe it, but I don't know for sure. But there is a good statistical argument to be made based on the size of the universe which I won't go into right now. Even if life is very unlikely indeed, we are probably not alone. However, almost the same argument suggests that we will probably never meet them. The same kind of argument cannot be made for god or talking cats. The only reasonable position on the alien question is some shade of agnostic. There is no justification for absolute certainty.
And just for the record, being emotional about an argument is different than basing a factual argument on your feelings. And even if, in the course of an argument, I do accuse the person of being irrational, well, the truth sometimes hurts.