The argument, at least in its modern form, focuses on two primary domains. Firstly, the complexity of living organisms and their origin, and secondly the origin of our universe. It is no coincidence that these two areas are both at the boundary of modern scientific understanding. You can trace religious apologetic arguments right back through time and notice how these arguments always ride on the boundary of the scientific understanding of the day, and fall out of favour as that boundary progresses outwards (i.e. as scientific understanding increases).
The argument from complexity is really a separate argument in its own right, and with its own flaws, but the way I approach it is more or less the same as with the various Fine Tuning arguments. I understand why these arguments are compelling to people, but I think they are compelling primarily due to subtle flaws or mistakes in our reasoning process.
Order Matters
The order in which arguments are presented seems to make a difference in how we perceive them. This shouldn't be the case if we were perfectly rational, but we're not. One of our more prevalent biases is called the "availability" bias. The more readily we can recall something, the more popular we assume it is. When reading through a series of arguments for some proposition, the first argument sets the tone, and then we tend to remember the final argument more than all of the preceding ones. Thus, if the final argument was weak, we may intuitively feel as though the whole proposition rested on a weak footing, regardless of how strong any of the previous arguments were.
I mention this because I think it plays a key role in how explanations for the Fine Tuning arguments are interpreted. Very often people will lay out their best arguments first, following up with more and more concessions, trying to show that even if we concede that the former arguments were defeated, the original Fine Tuning argument still has problems. However those later arguments are often weaker than the original one, and hence it can leave Fine Tuning seeming more favourable in the minds of the readers than if the arguments were listed in the reverse order. So that is what I'm going to do.
The Fine Tuning Argument
I am going to summarise the argument in broad terms so as to try to encompass as many variations of the argument as possible.
The Fine Tuning argument states that there are simply too many coincidences that need to have occurred in order for humans to have arisen, leading to the conclusion that our existence must have been brought about by the deliberate intent of an intelligent designer. The argument is most often made by people who believe in a particular deity, even though the argument itself typically doesn't make any direct statement about any particular qualities the deity must have, besides having the ability and desire to create a universe and life such as ours.
The Argument Isn't Falsifiable
The first problem I see with this argument is that it isn't predictive. It cannot be tested, because no matter what we see around us, one could always "explain" what we see by suggesting that "perhaps the deity simply wanted things to be that way". And with such hand-waving, any phenomenon no matter how absurd or obscure could be "explained" away. I say "explained" in quotes because such things are not actually explanations at all. They provide no additional information about the phenomenon, and it simply presents a brick wall preventing further discovery. You can ask "why" said deity wanted things to be that way and that might lead to some interesting conjecture, but there is no framework for coming to an answer about such questions. It's just guesswork all the way down, which shouldn't be so surprising because the argument itself is nothing more than a guess or a hunch, based on human intuition.
The more interesting question is "how" the deity might have gone about creating the phenomenon being observed, and in the process of discovering which processes were involved we might eventually realise that such processes could occur naturally and without any intelligent oversight. Such is the case throughout the history of science.
The Argument Doesn't Even Support Religion
Let's grant the argument in its entirety. Let's suppose that we have enough evidence to conclude that humanity must have been created by an intelligent deity. We exist because this being created the universe specifically to support life, and not just any life but specifically human life. The Earth was populated with humans because this deity specifically designed everything such that this would be so.
What does this tell us? Not much!
Based on the understanding that our universe was finely tuned for us to exist in it, we might conclude that there existed a being that had both the ability and the desire to create us. Great. Now what?
If believers want to connect the dots all the way to their holy book, they need to demonstrate how this particular deity must be the one mentioned in their book, how it couldn't be the one mentioned in any other book, and how it couldn't be a deity that never decided to write a book. That is, they still have all of their work cut out for them, in order to prove that any of their specific beliefs are true.
The Argument Doesn't Predict Intent
Suppose every claim of the Fine Tuning argument is true, but the intelligent designer created the universe and the earth with all of its abundant life forms, in an afternoon as a fun little side-project, and then left it to go and do something else. Or suppose the universe was birthed from the dying breath of such a deity, such that the deity no longer exists. Or suppose we live in a simulation, and human life is merely a side-effect of some larger project. Or suppose humanity was indeed the goal, but only to produce beings capable of producing energy that could be harnessed for other work.
All of these alternate hypotheses satisfy the Fine Tuning argument in every way, but they do not support any religious view. There is no hint of an afterlife, and such beings might require no worship or even any acknowledgement at all. They may not be interested in us, or they may have since died and ceased to exist. We could find evidence that conclusively proves the universe was finely tuned for life, but it wouldn't tell us anything about why it was created. Again, we are left with only pure speculation with no means to test it.
If the argument fails to support your religious views even when every aspect is satisfied, then it's probably not a good one.
We Don't Even Know Whether There Actually Was Any Fine Tuning
One popular argument for Fine Tuning claims that various parameters for the universe have values that are just right for life to arise and that any small changes to any one of these values would mean that life could not have arisen.
The problems with this have been widely publicised but in summary we don't actually know whether any of these parameters could even have had different values, and nor do we know anything about the process(es) by which such values might arise naturally. We don't know whether the values are special, whether they are contingent on other values, whether they can vary a little or a lot, or even at all!
Suppose the number we have for one of the values is an extremely small number. Why did we use those units? It's only a small number if we're using a larger scale. A distance of 1cm might seem small if we try to write that in Astronomical Units (the distance from Earth to the sun). But the same distance would be quite large at the atomic scale. If we don't know the scale, then we don't know how precise a value really is.
Further, why only change one value at a time? If we change multiple values together, we might produce life that is different than ours, but maybe it's still life nonetheless. Are we simply saying that "if the values were different, things would have been different"?
Why Was Fine Tuning Even Needed?
In the case of the parameters of the universe, an important question to ask is why any "fine tuning" was even needed in the first place. If we're talking about an all-powerful deity, why would such a deity be constrained by the strength of gravity and/or electromagnetism? Fine-tuning suggests the deity had limitations, which actually flies in the face of other claims made of deities.
This doesn't disprove the Fine Tuning argument at all. Rather it suggests the argument actually contradicts certain attributes for the proposed intelligent designer, namely omnipotence. If the Fine Tuning argument is true, then any intelligent designer must have had physical limitations or constraints placed upon it that were beyond its control.
There Are Alternative Explanations
The Earth orbits our sun at just the right distance for liquid water to form on its surface. This distance is known as the "goldilocks zone" because it's not too hot, and not too cold. Was that finely tuned? Let's think about it a bit more.
The goldilocks zone is not some narrow band but actually stretches quite some distance. Estimates for the habitable zone within the Solar System range from 0.38 to 10.0 Astronomical Units. It is thought that both Venus and Mars once also had liquid water. Further, the Earth's orbit isn't perfectly circular, meaning that its distance from the sun varies by several million kilometers.
Earth is in a very good location for liquid water to exist, but perhaps not the perfect location. Is it the only such planet in the universe? It might be, but there are good reasons to think it is not. We now know that most stars in our galaxy have a planet or planets orbiting them. Each galaxy contains approximately 100 million stars. And we know there are more than 2 trillion (that's 2 million million) galaxies in the universe according to the latest estimates. In fact the universe may actually be infinite, but we can only estimate the number of galaxies in the patch of the universe we can observe. That means there are a LOT of planets in the universe. Even if the probability of another Earth-like planet existing is extremely small, the sheer number of planets means there may well still be billions of planets very similar to Earth.
Why does Earth seem so special then? Perhaps it's because we are simply not used to thinking probabilistically in situations involving extremely large numbers of possibilities.
The Anthropic Principle
The probability of life arising on any one planet is indeed very unlikely, but given enough planets, we might still find that life is relatively common across the universe. Even if only 1 planet per galaxy contained life (an estimate far lower than most scientists predict), that's still potentially 2 trillion life-bearing planets!
But what about the universe? Well, most latest theories about the origin of the universe now involve some kind of multiverse. It may be that our universe is simply one of a potentially infinite sea of universes, each with different parameters. Perhaps the reason we find ourselves in this one is simply because this one has the particular parameters that allows for our kind of life to exist. It might be more surprising if we found ourselves in a universe where we shouldn't exist ;-)
Was History Finely Tuned?
Perhaps the best way to demonstrate the flaw in the Fine Tuning argument is to apply it to another scenario full of extremely unlikely coincidences, human history. We simply don't hear of anyone using the Fine Tuning argument to explain why all of the events in human history were specifically designed to result in their own birth. This argument seems absurd to most of us, but it's worth thinking about it because in doing so we can identify many of its flaws.
Imagine how many extremely unlikely events had to occur in the history of humanity in order for you to exist right now. We may feel uncomfortable with the idea that life on Earth is merely the product of a series of happy accidents, but when we look at all of the human events that led to our individual existence we do indeed find a very long series of happy accidents. The point of the exercise is to realise the sheer scale of the number of events that happen each day, and correlate that with the number of events happening across the universe on every planet, and perhaps across the multiverse as well!
Suppose history was indeed finely tuned. What would that mean for free will? Is our own life merely the fine-tuning for some future event? If you're like me you would feel very uncomfortable with such ideas, and yet somehow we don't feel the same about events in Earth's history, or events in the history of the universe. Why not?
Conclusion
The point I hope to have demonstrated is that our brains are not ideally suited to reasoning about the probability of particular events occurring in something as vast as the universe. We make inferences based on extremely small data sets, such as our personal experience over the last few minutes, or months, or even years for some. But none of this experience is sufficient to reason effectively about natural processes that take billions of years and occur a potentially infinite number of times throughout a potentially infinite multiverse.
It may be the case that our universe was finely tuned for life. I don't know. But the Fine Tuning argument as it is commonly presented isn't the way we'll find out. I find it uncompelling for the reasons I listed above. It's a guess based on incomplete data and flawed reasoning. We simply lack the data to really know for sure, and there are other explanations that would give the same outcomes that we observe today. The humble approach, for me, is to remain open to the possibilities and keep searching for more data.
And even if it turns out that the universe was finely tuned for life, I still see no connection with religion.