In “Does It Matter Whether Theism is Reasonable” (a short Chapter in the book, The Existence of God, 1965, Cornell University Press), Wallace Matson points out the difficulty involved in trying to rationally justify being rational:

Well, what can be said to someone who explicitly rejects reason? One might point out that very likely he does not reject it altogether; he wants evidence that the house he plans to buy is not infested with termites, he wants his sick children to be treated by competent physicians rather than by quacks, etc.; hence, he is inconsistent in accepting rational canons in all spheres of interest save one [i.e. concerning God or religious beliefs]. But this will not do, for the obvious retort is that consistency is a rational criterion, which is just what he is rejecting.
The situation is a queer one. It appears that there is no possibility of proving to the irrationalist that he should not be irrational, because any proof we might offer would, if cogent at all, presuppose canons of logic and evidence, and in consequence would be circular. You cannot checkmate a man who refuses to play chess. (The Existence of God, p. 242-243).

This is a recurring theme in epistemology: the problem of trying to justify basic principles of logic and evidence without making use of the very same principle(s) in the justification. The long-standing problem of induction, for example, might be stated as the problem of justifying inductive reasoning without begging the question by making use of inductive reasoning in the justification.

There is a real puzzle or difficulty that Matson is pointing out, but the above statement is too quick and dirty to draw the conclusion that there is no hope of rationally justifying being rational.

First, there is vagueness and unclarity in the term “consistency” that needs to be fixed. One might reasonably use different methodologies and criteria depending on the subject matter that one is dealing with. The scientific method may not be appropriate to apply to historical questions or to philosophical questions. Criteria for evaluation of mathematical or logical proofs might not be adequate for dealing with scientific questions. Historical and scientific methods might not be adequate for dealing with moral issues. Use of different methods and criteria for evaluation of claims and theories does not appear to involve logical inconsistency. Different subject matters may require different methods and criteria.

On the other hand, if logical consistency refers to the fundamental principle of non-contradiction, the idea that logical contradiction is to be avoided, then there really is no alternative available. Whether one is trying to figure out an issue in history, science, philosophy, or religion, logical contradiction must be avoided for the sake of meaning and coherence. If the claim “God exists” does not rule out “It is not the case that God exists”, then there is no coherent meaning to “God exists” (unless you reduce this to a purely subjective claim about the feelings of the speaker, e.g. “I have positive and hopeful feelings about the universe”.) If the claim “Jesus rose from the dead” does not rule out “Jesus died and never came back to life.” then the former claim has no coherent meaning. Once you allow logical contradictions in an area of thinking, meaning and coherence are eliminated.

This sort of logical consistency (avoiding logical contradictions) is a rational criterion that is accepted by the irrationalist in most areas of thought, otherwise no meaningful and coherent communication would be possible. So, the irrationalist does accept this criterion of rationality, and it is reasonable to point this fact out to an irrationalist. Furthermore, if the irrationalist insists on discarding this principle in the area of religious belief, you can point out how this leads to incoherence and the destruction of meaning in relation to religious claims or beliefs.