Okay, this isn't an irrationality that relates directly to organized religion, in fact, it can often come about more in the minds of those who reject notions of god, afterlife, special meaning to life, etc. I present it, however, because it is an irrationality that can prove counter-productive to one's case against these aforementioned concepts. Better to have one's position rooted solidly....
The idea is that we're just chemicals, we're just the result of these complex chemical reactions, that things like our identity and our feelings are just the end result of electrochemical processes. The reason this is illogical is the qualifying word "just". As if something was being diminished by this being the case. But there's no basis in reality to think that these facts subtract something. They don't.
The fact that unsubstantiated beliefs are unsubstantiated is enough to discredit holding these beliefs. Acting as if the chemical and biological realities of existence somehow subtract away value or meaning from life doesn't accomplish anything, and it is itself unsubstantiated, as there isn't actually anything else with which to make the comparison.