My suspicion is that many theists don't think atheist are sincere when we ask for evidence because we reject the evidence they do give us. If we then modify their statements to read "It frustrates us theists when you atheists keep asking for evidence and then reject the evidence we give you," and "You atheists need to sincerely investigate," we remove the contradiction. The theist, apparently dumbfounded by how the atheist could possibly reject evidence that they find so obviously points to their god, concludes that the atheist rejects their evidence because the atheist is closed-minded.
The error that the theist makes, though, is that we are not rejecting this evidence straight out of hand. If we were, that would be closed-minded. Instead, we do take a look at it and often find it quite inadequate. In fact, we are sometimes baffled at how theists think their evidence is so great. Here are just some examples of what I mean (some of these address religion more in general):
- Subjective experiences — I've already discussed why this is unacceptable. The inconsistencies in these experiences show them to be unreliable. One thing I would add that I did not say there is that the theist might object claiming there are still a lot of similarities between these experience, so in general they are the same and we shouldn't get picky over the trivial details. To that I would say, in perhaps a way the theist will understand, "the devil is in the details!"
- Miraculous healing — I briefly mentioned this before. The story was about a theist's mother having a deformed foot that three doctors had told her would need surgery. Yet she claims God healed her foot for her without surgery. Additionally, the theist made note that we might argue that all three doctors were wrong in their diagnosis and asked if we could really believe that. That answer is yes. First of all, we know doctors have made false diagnoses in the past. Second, I recall Neil deGrasse Tyson once making a point that doctors basically all come from the same school of thought, so you're not actually getting three separate opinions; you're getting the same opinion three times. Therefore, the probability of all three doctors making a false diagnosis is nearly the same as one doctor making a false diagnosis...or at least the probability isn't as low as the theist thinks it is. Third, we know that memory is volatile and changes to fit a narrative. Did this theist's mother really get those diagnoses? Or has her memory constructed these to fit a narrative? Or is it the theist's memory that is constructed to fit a narrative (in other words, did the theists mother ever claim those doctors made those diagnoses)? Fourth, even if we assume the theist is correct in their memory, there are still other people who have experienced what they claim to be miraculous healings and they attribute it to a different god. By which I mean the credit that God is getting is subjective, so this has to be discarded. Therefore, the cause is actually unknown. To attribute it to God would be committing the argument from ignorance fallacy. Fifth: confirmation bias and unanswered questions. When a supposed miraculous healing occurs (for the sake of argument, let's grant that the theist's god is performing these miracles), we often hear from the religious how great and awesome their god is. Then what about all the religious people out there who pray for healing and don't get healed, even those who perhaps have similar foot deformations to what this theist's mother supposedly had? The theists seem to forget about those people when considering how great their god is. The primary point here is that the personality characteristics these theists give their god do not align with reality as it seems this god answers prayer based on some sort of lottery system, and how would that be any different if there was no god and things just happen by random chance? (Notice that this problem also has elements of subjective experience to it.)
- Eyewitness testimony — This relates to those who supposedly saw Jesus perform miracles or saw the resurrected Jesus shortly after he supposedly died (if he ever lived). I remember one Christian friend once claiming there were "thousands" of eyewitnesses. At that time, I was not familiar with such arguments and didn't know how to respond. When I looked into things, there just isn't enough evidence to support that claim. What we have is a few books (books that are part of the New Testament) that claim there were many witnesses. If we are to believe there were all these eyewitnesses, we have to believe the books. But is there any reason to trust these books? The short answer to this is "No." Some things that could help is if there were more writings from people who were supposedly witnesses and (even better) Roman documentation of some of these events. For the former, the Christians will bring up the fact that a lot of people were illiterate. So what? Absence of evidence is still absence of evidence, even if we have a reason to expect an absence. All the Christian's argument does is prevent this absence of evidence from becoming evidence of absence. (If there is no evidence for a claim and we expect that there should be, this lack of evidence actually becomes evidence that the claim is false. In this case, we don't expect there to be evidence, so that fact that we lack evidence gets us nowhere.) So, I grant that there may have been thousands of witnesses, but we just don't have enough evidence to accept such a claim (or reach any conclusion, one way or another). I hope I need not reexplain why we can't just trust that the authors were truthful and/or correct. (Note that this is assuming that eyewitness testimony is a valuable form of evidence. The reality is that it is not. So even if we actually did have a bunch of eyewitness testimony, it would not matter anyway.)
- Empty tomb - This is about how do we explain that Jesus' tomb was empty. This has the same basic problems as the example above because the idea is based on eyewitness testimony. Once again, are the stories reliable? Looking at the contradictions in the these stories, the short answer is "No." (Was the stone already rolled back when people arrived? Who were these people? What did they see inside and outside the tomb? For those stories that say people saw Jesus, where did Jesus say he was going? And so forth. All of these have different answers depending on the gospel.) Once again, I have seen a bizarre defense from Christians — their claim is that we should expect contradictions. Once again, they are correct, but again their point is...pointless! The whole point seems to be that since human testimony is flawed and usually contains mistakes and since this testimony is flawed and contains mistakes, we can trust it! As an engineer, such rationalizations (and it scares me when engineers make such rationalizations) are worrisome. I work in a field where the goal is to eliminate mistakes, because when mistakes happen, bad things are likely to follow. To hear people suggest that mistakes are acceptable goes against the philosophy of my profession. (And I think Christians who expect the products they buy to be engineered well should consider this when wondering why atheists aren't buying their product.)
- Die for a lie? — This question is asked about Jesus' disciples. Why would they risk persecution and death if they knew they were just making stuff up? There are multiple problems with this. First, it creates a false dichotomy — either the disciples where lying or they were telling the truth. Why couldn't they be deceived, deluded, and/or mistaken? (I have seen some apologists address this, but give weak evidence to rule out this option.) There is a reason to create such a dichotomy, because there are examples of other religions that faced and flourished under persecution, like Mormonism. Since their religion cannot be true (since the Christian thinks their religion is true and only one of them can be true), the early Mormons must have been deceived, and that is why they would risk persecution for what others see as a lie. The second is context. As much as Christians like to claim atheists take Bible verses out of context, this is a place where being out of context works in their favor. See, it seems that one of the ideas behind this question is to put the person being asked in the shoes of the disciples. But I see a lot of people failing to do this. Take the idea of risking persecution. The counter idea to this that is implied is that life would be just peachy if they remained Jews. When you look at what the Romans thought about Jews, this is just not the case. There was going to be oppression either way, so persecution doesn't seem to be that big of a factor. Even if life would have been peachy as a Jew, we can still go point to Mormonism. They wouldn't have had to worry about persecution if they had remained Christians (as many of them likely were before converting), so why convert? In short, I think a lot of people go about thinking "Well, I wouldn't convert!" and fail to consider that other people would.