There is a big difference between falsifiable and false. The theory of gravity is falsifiable, but not false. If two bodies of mass were shown to be unaffected by the force of gravity, then the theory would be proven false. Thus, there is a conceivable method for demonstrating the theory to be false, if such evidence could be found. But, no such evidence has been found, and thus the theory of gravity is given the label of "theory", which is essentially the scientific seal of approval that it is generally accepted as true.
Most people who believe in religion will say things to the effect that their beliefs are unfalsifiable, although not commonly using that term. Usually it's worded something "nothing will make my faith waver" or "I know beyond a shadow of a doubt". I wish to discuss why this is harmful. As the picture above states, if your belief is not affected by reality then it is not based on reality. That is, if your belief is unfalsifiable then there is no rational reason for you to believe it.
I wish to give an example of how belief in god is unfalsifiable. First, I will give an example of how a scientific experiment works and contrast that with the types of experiments believers typically use. I wish to test whether a certain kind of medicine is effective at curing a certain disease. My hypothesis is that it does. The hypothesis is falsifiable because I can observe the bacteria counts in the subject's bloodstream. If the counts remain the same after the medicine is administered, then the conclusion must be drawn that the medicine is ineffective and thus the hypothesis is falsified. So, I would proceed to give the medicine to multiple infected subjects and measure their symptoms and determine whether the medicine was effective or not. There are (at least) two possible outcomes--one which would indicate that the medicine is effective and one which would indicate that it is not.
Now, let us perform the same experiment with testing the existence of a god. Suppose that a person is inflicted with an illness. I wish to determine whether my god exists. I pray to my god and ask it to heal the person who is ill. If the person recovers, I attribute the recovery to my god and say that my prayers have been answered. If the person does not recover, I attribute the lack of recovery to a demonstration of god's will, concluding that my god has some higher purpose for the person or that it is somehow part of its plan to keep my friend ill. Either way, the conclusion is that my god still exists. That is, I have already made up my mind that my god exists and regardless of any evidence presented, I will continue to believe. In fact, this is the case with any unfalsifiable claim. If you adopt the claim to begin with, you will continue to believe it regardless of any evidence presented.
Now, this discussion should not be construed to mean that I believe I am presenting evidence that there is no god. I wouldn't do that. I'm claiming that it is an unfalsifiable claim, and therefore I am admitting that I cannot prove non-existence. A prayer going unanswered is not proof that no gods exist. What I am saying here is that there is a difference between a falsifiable scientific hypothesis and a belief, accepted a priori, which is unfalsifiable.
To me, this is a very personal matter. My own wife died of cancer three and a half years ago. She was diagnosed seven years prior to her death. She had been through multiple rounds of chemotherapy, radiation, and even two different trial drugs that were in the stages of research. Ultimately, all of the treatment was not sufficient to remove the cancer from her body and it took her life. During these seven years, her family and so many of her loved ones (including myself) fasted and prayed for her countless times. She received special priesthood blessings from her father, uncles, and me. If she had lived, it would be chalked up to a miracle and the credit would be given to god (even though the credit would rightly belong to all the years of scientific research that have developed the cancer treatments that we now have). Her death was accepted by all of us (myself included) as the will of god. We believed simply that she was more needed in Heaven than she was here on the Earth, that her work here was done.
The point is that regardless of the outcome, the conclusion is the same: that god lives. This is the case with every single possible test that could be presented to determine whether there is a god. The Book of Mormon contains a "promise" near the end of it that asks the reader to ponder the things ey has read, to pray about it and ask god whether the things in the book are true. If a warm fuzzy feeling is felt, that is supposed to be the Holy Ghost indicating that the book really is true. However, this test is as well unfalsifiable. If no feeling is felt, then the explanation is one of several things--that something was not present in the formula for making the feeling come. Perhaps you didn't pray sincerely enough. Perhaps you didn't read enough of the book or didn't understand it fully. Perhaps you didn't have enough faith when you prayed. The conclusion is never "Perhaps the book is not true." The only two possibilities are that it is true or that you did something wrong. Thus, this test is rendered completely useless. The only acceptable outcome is "it worked". If I were to present a hypothesis such as this one, design an experiment around it, and try to publish a paper on the matter, I would be laughed out of every scientific journal in the world because I have assumed what I'm trying to prove in the hypothesis itself.
The same is true in mathematics. In fact, in mathematics, we only talk about statements that are called "truth statements". Loosely speaking, a truth statement is a statement that is either true or false. If a statement is not of this nature, then it is not discussed at all in the mathematical world. An example of a statement that cannot be shown to be true or false is "This sentence is false." It is a logical paradox, and therefore it is not considered in rational discussion. But even if a statement can be proven either true or false, such as "every integer greater than 2 can be expressed as the sum of two prime numbers", it is not accepted as true until it is proven. The statement here is known as Goldbach's Conjecture and has not yet been proven true, thus it remains a conjecture. It is falsifiable because if a number greater than 2 is found that cannot be written as the sum of two primes, then such a number becomes a counterexample and the conjecture is shown to be false. No such number has yet been found and computers have tested numbers up to 1018. A math paper may say something along the lines of "If Goldbach's Conjecture holds, then..." but no mathematician would claim that it is in fact true without a rigorous proof indicating why it is true.
The point is that there is no rational justification for belief in any unfalsifiable claim. If there is no method to prove your belief wrong then you must accept it without reason. Bertrand Russell very wisely said "If something is true, then you should believe it. If it is false, then you should not believe it. And if it cannot be determined whether it is true or false, then you should withhold judgement."
The statement "there is a god" is just vague enough that it is unfalsifiable. However, once attributes are given to this god, some of them may be falsifiable (and in any case I'm aware of, also false). For example, the claim that god answers every prayer. This is falsifiable, and also false. In fact, there are many cases where it would be impossible for god to answer everyone's prayer since different people may be praying for different things (eg, a farmer praying for rain and a tourist praying for fair weather on the same day). To give the apologetic rebuttal that god answers prayers in his own way would then make the claim unfalsifiable, since there is no way to prove that a given outcome is evidence that god did not answer the prayer. But, if the original meaning is to be understood, then the claim is clearly false.
What's the benefit of ensuring that claims are falsifiable before believing them? Because humans are gullible. There's a reason there's such a big market for homeopathy, penis enlargement pills and devices, and many other things. We humans are susceptible to false beliefs. Many guys fantasize about having a larger penis. The idea is appealing. So, belief that it is possible by ingesting certain herbs is appealing. Of course, these claims can be falsified. So, it is important to check these claims, see if any research has been done to indicate that they are true, and determine whether the investment is worth it or simply a con.
Thus it is important to be able to answer the question "If I'm wrong, is there a way to know it?" If there is a way to know whether you're wrong, test it and see if you really are wrong. If there is no possible way to know whether you're wrong, then why are you believing the thing that you believe? There is no evidence to prove that mediums do not talk to the dead, so if you believe that they do you'll be endlessly trapped in that belief and be taken advantage of by mediums who charge you high prices for reading sessions where they pretend to tell you about that hidden stash of money in Uncle Vernon's chimney. Just the other day, I saw a friend post a story about a family who prevented themselves from getting the flu by having onions in every room of the house. Allegedly, the onions attracted all of the flu viruses, so the family didn't get it. This is a falsifiable claim, so let's go about deciding whether it's really true or false. And if something is unfalsifiable, let's set it aside and not worry about it.