Perhaps some fallacies that religions or their adherents tend to use:
1. Faulty Cause: (post hoc ergo propter hoc) mistakes correlation or association for causation, by assuming that because one thing follows another it was caused by the other.
Perhaps commonly seen in such things as so-called miracles, prayer, good behaviour because one is religious, etc.
2. Appeal to Ignorance: (argumentum ad ignorantiam) attempts to use an opponent's inability to disprove a conclusion as proof of the validity of the conclusion, i.e. "You can't prove I'm wrong, so I must be right."
Seems that many religions rely on this - we can't prove that much of what they espouse is wrong, even down to the most basic belief - that God exists.
3. Bifurcation: (either-or, black or white, all or nothing fallacy) assumes that two categories are mutually exclusive and exhaustive, that is, something is either a member of one or the other, but not both or some third category.
Perhaps best seen by those believing that if one doesn't subscribe to their particular faith and beliefs, one is doomed to some nasty end, unlike them.
4. False Dilemma: (a form of bifurcation) implies that one of two outcomes is inevitable, and both have negative consequences.
As above, deeming that those not of the faith are doomed.
5. Faulty Sign: (also includes argument from circumstance) wrongly assumes that one event or phenomenon is a reliable indicator or predictor of another event or phenomenon.
Perhaps more so in the past than the present, when events were often tied to bad behaviour and such.
6. Damning the Source: (ad hominem, sometimes called the genetic fallacy) attempts to refute an argument by indicting the source of the argument, rather than the substance of the argument itself.
Obviously all atheists are wrong, and more the reverse of this, that is, that the truth only comes from the religious teachings.
7. Tautology: (a sub-category of circular argument) defining terms or qualifying an argument in such a way that it would be impossible to disprove the argument. Often, the rationale for the argument is merely a restatement of the conclusion in different words.
If anyone hasn't come across, 'because the Bible tells us so' in debates with Christians, then they haven't been around long.
8. Appeal to Authority: (ipse dixit also called ad verecundiam sometimes) attempts to justify an argument by citing a highly admired or well-known (but not necessarily qualified) figure who supports the conclusion being offered.
Much the same here, but applies to all the interpreters of religious writing too, who often just recycle the original material.
9. Appeal to Tradition: (don't rock the boat or ad verecundiam) based on the principle of "letting sleeping dogs lie". We should continue to do things as they have been done in the past. We shouldn't challenge time-honored customs or traditions.
Religion wouldn't be religion without this of course.
10. Appeal to the Crowd: (ad populum or playing to the gallery) refers to popular opinion or majority sentiment in order to provide support for a claim. Often the "common man" or "common sense" provides the basis for the claim.
Commonly used by those arguing about religious beliefs - the majority (who tend to have a religious belief) just cannot be wrong, even though they are split as to what belief they actually might have.
11. Slippery Slope: (sometimes called a snowball argument or domino theory) suggests that if one step or action is taken it will invariably lead to similar steps or actions, the end results of which are negative or undesirable. A slippery slope always assume a chain reaction of cause-effect events which result in some eventual dire outcome.
Commonly used concerning morals - where it is wrongfully assumed that, firstly, morals originated with religions (which they undoubtedly didn't), and secondly, that people would lack morals if they had no such religious beliefs to guide them. This is refuted by the fact that there is good evidence that the non-religious are just as, or more moral, than those who do have a religious belief.
12. Appealing to Extremes: A fallacy very similar to slippery slope, which involves taking an argumentative claim or assertion to its extreme, even though the arguer does not advocate the extreme interpretation. The difference between the two fallacies is that appealing to extremes does not necessarily involve a sequence of causal connections.
Perhaps seen best by those claiming that without a religious belief, life would have no meaning.
13. Hypothesis Contrary to Fact: This fallacy consists of offering a poorly supported claim about what might have happened in the past or future if circumstances or conditions were other than they actually were or are. The fallacy also involves treating hypothetical situations as if they were fact.
As above, in that all sorts of calamities are envisaged if religions fell into disrepute - which is often why we do have religious apologists squealing away like stuck pigs every time they feel threatened or where they see religious influence disappearing. Witness the bile from Giles Fraser when John Humphrys commented about the privileged place for Thought for The Day on Radio 4. (UK)
14. Non Sequitur: (literally means "does not follow") in a general sense any argument which fails to establish a connection between the premises and the conclusion may be called a non-sequitar. In practice, however, the label non-sequitar tends to be reserved for arguments in which irrelevant reasons are offered to support a claim.
Perhaps the rise of civilisation might be cited here, where religions are often seen as, or claim to be, the only driving force causing this.
15. Red Herring: attempting to hide a weakness in an argument by drawing attention away from the real issue. A red herring fallacy is thus a diversionary tactic or an attempt to confuse or fog the issue being debated. The name of the fallacy comes from the days of fox hunting, when a herring was dragged across the trail of a fox in order to throw the dogs off the scent.
Perhaps citing all the good that religions have done and disregarding any bad things that have happened.
16. Inconsistency: advancing an argument that is self-contradictory, or that is based on mutually inconsistent premises.
As in, “Their religious belief doesn't make sense but ours does!”
17. Appeal to Probability: a statement that takes something for granted because it would probably be the case (or might be the case).
Like the fact that life, and humans evolving, is just so improbable without some divine creator - and management team.
18. Argument From Incredulity (appeal to common sense): "I cannot imagine how this could be true; therefore, it must be false."
As above about the probability of life coming into existence.
19. Appeal to Emotion: an argument is made due to the manipulation of emotions, rather than the use of valid reasoning.
This is most obviously played upon by the religious in the way that such things as an afterlife are posited. Nice to think that things are better when we die even if this isn't necessarily true.
20. Appeal to Fear: an argument is made by increasing fear and prejudice towards the opposing side.
The opposite of the above - no enticements, but the stick instead, as in Hell.