Is faith rational? Yes. No. Sometimes. Not always. The answer depends on the question. Which faith are we talking about, and what do we mean by rational?
Ask an atheist and you're likely to hear that faith is always "believing what you know isn't true," or "belief without evidence." As for rational, it's common to hear atheists describe rationality in terms of believing only what can be demonstrated by empirical methods, preferably science.
If those definitions were correct, the debate would be over. End of story; faith is irrational. But there's something odd about the way these definitions appear so conveniently to make the atheists' point about faith, without needing to look at any actual faith as evidence! This should at least give a thinking person reason to pause and wonder.
In fact, neither definition is good, faith nor rationality. The game isn't up; the debate isn't over. Christian faith — the only kind of faith I care to defend; other forms may indeed be irrational, as far as I'm concerned — has stood on rational foundations from the beginning.
It's certainly been tied to evidence from the start. Jesus Christ "presented himself alive… by many convincing proofs" (Acts 1:2, ESV). In Acts 17 Paul reasons from various sources for the truth of the resurrection. Those are just two evidence-oriented examples, out of dozens if not hundreds in the Bible. Whether or not one agrees that the Bible is inspired by God, it's certainly been the source from which Christians — in fact all of Western civilization — has always derived its understanding of faith. Therefore, any definition that contradicts this understanding of the word also contradicts its long historical usage. Christians have always understood faith to be connected to reason and evidences.
For that reason, it's also no new invention for Christians to use reason and evidence to support our faith. Apologetics began in the New Testament and it's been practiced continuously ever since. If faith is irrational, then Augustine, Aquinas, and Pascal were irrational. So was Galileo, who (contrary to myth) was a believer in Christ; and also Kepler, Brahe, Newton, Faraday, Maxwell, Linnaeus, Mendel, and von Braun, all of them believers, to name just a few from among hundreds of history's brightest lights. While not all of them were apologists and thus not all left a record of how they connected faith to reason, hundreds did, including many who are counted among the greats of Western intellectual history. The modern evidence-free atheist view of faith is pure mischaracterization.
There are problems in atheists' view of reason as well. Survey the early New Atheist literature (from Harris's The End of Faith through at least Hitchens's God Is Not Great) and you'll find rationality or reason described in terms of conforming one's beliefs to objective, empirically provable evidence. This view is quite ironically irrational. (I've seen better depictions of reason in atheist literature since then, but they've been the exception, not the rule.)
It's irrational first because it trips over its own feet. If there's a rule that says no belief is rational unless it can be demonstrated empirically, then the first thing the rule tells us to do is to discard the rule. Its truth can't be demonstrated empirically.
Second, we all know many things to be true without empirical proof. Two examples: First, when my wife and I first spoke our love to each other, there was no empirical proof our words were true, yet we both knew it. Second, the rules of logic can't be demonstrated empirically.
Third (and worst), this rule implies that if there's a God, he can't make himself known except by empirical means. This is a breathtakingly dogmatic theological assumption, coming as it does from people who deny any knowledge of theology: "If there's a God, the only way we could know it would be if he made himself known through scientifically discernible means." No. If a Creator God exists and wants to make himself known, atheists won't be setting his rules for him. An all-powerful, all-wise God can reveal himself by any means he chooses.
Which he has done. God has made himself known through the glory of his creation; through the uniqueness of human nature, including our nature-transcending capacities for rationality and free will; and above all through the resurrection of Jesus Christ, for which considerable evidence does in fact exist.
Space doesn't allow me to repeat all that we know about these evidences, which are enough to fill whole libraries. Instead — since it's quicker and just as relevant — I want to turn the original question on its head: Is disbelief rational?