Fundamentalist atheists are atheists who are dogmatic in their rejection of religion. They are so sure they are right that they don't care about research into the nature or effects of religion.
Fundamentalist atheism cannot exist because there are no "fundamental" beliefs for atheists to hold. This myth demonstrates why this is true by attempting to create out of thin air a belief for atheists: the idea that they reject religion so firmly that they no longer care about further research on it and are therefore dogmatic in their views. The errors made in this myth are so simple, basic, and obvious that it's difficult to credit anyone who repeats it with understanding atheism at all.
It is true that there are atheists who might be described as "dogmatic" in their rejection of religion they are convinced that the impact of religion is more negative than positive on balance and aren't interested in further research on religion. The fact that such atheists exist, however, cannot automatically justify the decision to label them "fundamentalist" atheists and the reason why lies in the nature of atheism and fundamentalism.
For one thing, not even a vaguely negative view of religion is inherent or basic to atheism itself, much less a dogmatically negative view of religion. Some atheists are themselves religious, following religions where gods are either unnecessary or even specifically rejected. Other atheists have a more sympathetic view towards religion and doubt that it's effects are ultimately more negative than positive. Still other atheists do reject religions as evil, but don't necessarily feel that they have nothing to learn from further research.
Fundamentalism is a label applied to religious movements that, at the very least, emphasize the importance of "fundamental" beliefs in contrast to modern developments. Atheism isn't even a philosophy or a belief system, much less a religion, but even if we ignore that we still have to conclude that "fundamentalist" can't apply here because the rejection of religion isn't a "fundamental" belief.
Atheists who reject religion in the manner described by this myth aren't arguing that atheists need to agree with them in order to "get back" to the original basis of some "atheistic faith." They are, instead, making a social and political argument which they would like both atheists and theists to agree with just like anyone who makes an argument for any social or political position. Is an atheist who is "dogmatic" about using logic a "fundamentalist" atheist? No, because logic isn't a "fundamental" belief in atheism. Is an atheist who is "dogmatic" about reading widely a "fundamentalist" atheist? No, because that also isn't a "fundamental" belief in atheism.
It's unlikely that such atheists would be described as "fundamentalists," despite their dogmatism. If I'm right about this, then it suggests that it isn't the dogmatism per se which outrages critics of atheism, but simply the fact that atheists argue that religion is more harmful than beneficial. Labeling atheists making this argument as "fundamentalists" is an effective way to encourage others to dismiss or ignore those arguments without ever directly addressing them, much less rebutting them. It's a form of ad hominem argument known as the Genetic Fallacy: don't believe or listen to these people because they are fundamentalists.
Another important problem with this claim about fundamentalist atheists is the general absence of any supporting documentation. If you see people repeating it, chances are you won't find them providing any hard data of atheists specifically dismissing the value of any further research on religion or specifically rejecting the possibility that research might prove them wrong. You also won't find them providing any evidence atheists being "dogmatic." There are atheists who don't think it likely that they will be proven wrong, but that isn't dogmatism. Most scientists don't think that basic principles of gravity or evolution will be proven wrong, either, but that doesn't make them dogmatic.
Even if such atheists are mistaken, they can't justifiably be accused of dogmatism if they sincerely think that they have an abundance of hard data supporting their position such that new data is unlike to contradict all that's come before. Further research into religion might be interesting, but it's inappropriate to insist that atheist critics of religion don't care much about such research without being able to show that there are good reasons to think that this research will have an impact on atheists' conclusions.
There is also a real double-standard here in that irreligious atheists who are critical of religion are expected to "moderate" their negative conclusions about religion, but you don't see similar statements about religious theists who think religion is a good thing. Why aren't theists "dogmatic" for insisting that religion is necessarily a good thing and who are not interested in further research before latching on to this conclusions firmly? Why aren't Christians called "fundamentalists" when they insist that Christianity is a force for good and without wondering if further research will prove them right?
So many religious theists "think they already know, without study" that religion is a good thing, necessary for morality, necessary for good government, etc., but I don't see them being accused of being fundamentalists because of it. Once again, this suggests that the problem isn't the alleged dogmatism, but the atheism itself and the sharp criticism of religion. Thus we find that accusations of dogmatism and fundamentalism are just a mask for attacking atheism and encouraging people to be dismissive of atheistic critiques of religion without ever doing any real work to address those critiques. It's dishonesty being given a pseudo-academic mask of respectability.