1. Materialism. This is the view that everything in the universe is made of or comes from matter or energy. This view must be automatically assumed by atheists, but it is a view that is widely debated philosophically. I would even go so far to say that the creation of the universe must depend on a nonmaterial source, making materialism an incorrect and logically indefensible position (more on this later). Initially, it may seem as though holding a materialistic view might be normal and not up for debate, but here are some of the implications of it, from the impossible to the hardly controversial.
A. Origins. The most difficult and what I see as most irrational belief one must hold to maintain a atheistic/materialistic worldview is the belief that the universe was created out of nothing. If God did not create the universe, then there are only two options to choose from regarding its creation. The first one is that it created itself...out of nothing. No matter, energy, light, or time. NOTHING. The impossibility of this should be apparent. If nothing exists, then there is nothing there to create something, and no existence of time in which something could change from nothing into something.There are certainly variations between atheists on the specifics of these beliefs, but the general views presented here are absolutely necessary and fundamental for anyone who claims to be an atheist and wants to be consistent in their beliefs.
The second option is a little more specious, but equally impossible, and that is that the universe is not created, but has always existed. This argument doesn't deny the Big Bang, only suggests that the Big Bang was caused by another universe (or other physical cause), which was created by another universe, and so on ad infinitum. There is a logical and physical flaw with this view. If time has always existed, then there was been an infinite amount of time between now and the beginning, which means now would never have arrived. It would be similar to counting on a timeline from negative infinite to zero (if now equals zero) or any number really. The other problem is that with an infinite amount of time, all the energy in the universe(s) would basically run out (second law of thermodynamics).
Finally, there is the ridiculousness that would occur with probabilities. If there were an infinite amount of time, then everything that could possibly happen will and has happened...an infinite number of times. For instance, if the probability that a universe has the conditions necessary to sustain life is 1 out of 100 (although people believe its much less than that, more like 1 out of a trillion or more, but that is not important for now) then the probability that a non-life sustaining universe is 99 out of 100. So this means that there will be 99 non-life sustaining universes to every one that has life; however, if time is infinite, then there is an infinite number of life sustaining universes and an infinite number of non-life sustaining universes. There is an infinite number of both, but somehow, there have been more non-life sustaining universes than life sustaining universes. Still more ridiculous is that there is the possibility of going against the odds. This means that it is possible that there are more life-sustaining universes that have been created than non-life sustaining universes and now there are two contradictory possibilities that must be true at any given point in time, but only one can be.
B. Evolution. I am not arguing for or against evolution, but merely pointing out that if a person holds a materialistic view of the world, they must accept evolution has occurred and is a completely naturalistic process. No divine guidance or help at any point in time. For many, this is not a problem, but when you start to investigate the problems, gaps, and inconsistencies in evolutionary theory, it becomes clear that many people (e.g. Richard Dawkins) who so vigorously defend evolution, are doing so in an attempt to defend their worldview rather than a scientific theory. In other words, the materialists who claim to be unbiased or "free" thinkers, have biases just like everyone else.
C. Supernatural. There exists many beliefs regarding supernatural beings or things. Some accepted by Christians and some not. Other than God, other supernatural forces that people believe in include ghosts, karma, precognition or extra sensory perception, angels, demons, other spirits, rebirth or reincarnation, astrology, fortune telling, and the human soul. Many atheists do not have a problem not believing in these things, but others strongly cling to one or some of these, especially the idea that people have a soul. If there is no God, there is no logical explanation for any of these supernatural beings or phenomena, making it irrational and inconsistent to believe in even one of these things if God's existence is denied.