It's not self-refuting.
I'll address some justifications I've seen people give for why it is.
It's self-referential, as in consciousness attempts to explain itself
If being self-referential in itself is deemed to be a problem, this would be asserting that consciousness is fundamentally inexplicable, as any attempt at an explanation would suffer from the same problem. Anyone taking such a position would not be able to make any claim whatsoever about the nature of consciousness (whether natural or not), and may be precluded from making any statement whatsoever about any system they're a part of, like saying how fast a car is going, when you're inside the car, or making any claims about your organs or how your body works, at least or especially in as far as it relates to the brain.
This seems to necessitate anyone who has a problem with it being self-referential to reject all of neuroscience and probably psychology, at the very least, because those explicitly involve trying to explain consciousness.
This doesn't actually seem to be a problem at all.
If there were only one single consciousness in existence, I might perhaps agree that we'd be a lot more limited in our ability to explain it (e.g. studying the emergence of consciousness would be out the window). But supposing that you accept the existence of other minds, that allows for a whole lot of potential to study consciousness in a way that doesn't include yourself (although reflecting on your own thoughts does provide a lot of useful insight into what your conscious mind is actually doing, and your own consciousness is the only one you can experience first-hand).
Can reason justify reason?
More specifically, people seem to have an issue with using reason (or logic) to explain or justify reason. But reason is justified by principles of reasoning, logical frameworks, philosophy, and so forth, as well as by the fact that reason has shown to be reliable in corresponding to reality (as best we can tell, and to a degree that has proven to be extremely useful in helping us thrive in the world).
So this wouldn't be self-refuting, because this justification is entirely distinct from how we explain the underlying mechanism of our brains that allows us to reason.
If our thoughts are deterministic, can we really think?
There's also a question of determinism, in that if our thoughts are deterministic, can we really think or reason? And the answer is "yes". ... Or "no", if you've snuck non-determinism into your definition of "think".
But, bottom line: there's nothing in determinism that would prevent us from going through some steps in order to reach a conclusion that depends on certain criteria (which one might reasonably refer to as "think"). If you've ever written some non-trivial code, you'd know that deterministic computers can already do this (even though they probably don't yet meet the definition of being "conscious").
Brain stuff is non-physical?
There also seems to be some vagueness of implicitly asserting that ideas and consciousness and whatnot are non-physical, and then working backwards from there to say that naturalism can't explain that. This wouldn't be self-refuting, but this is also begging the question: assuming the thing you're trying to prove.
Ideas exist within our minds, and consciousness is a part of our minds, so this would be explained as part of explaining how our minds work.
Perception explaining perception?
Some may also take issue with using perception to explain perception, which, if you think that's an issue, seems to lead to solipsism (concluding that other minds and/or external reality don't exist, or that we can't be sure of their existence). But this doesn't seem self-refuting. It's just that we cannot fully prove that our perception reflects reality, but that seem to be the most likely and reasonable conclusion. Failing that, we can also just say that we seem to be observing some form of reality within which we seem to be operating, and that would be sufficient to build a model of what is being perceived, and use that to inform our actions (regardless of whether or not that is "real").
"Eat food to not starve" is something that seems to be followed quite universally across the human race, regardless of beliefs about naturalism or solipsism or whatnot.
Can naturalism account for the experience of consciousness?
People also say that naturalism cannot account for the subjective, first-person experience of consciousness, but this wouldn't make it self-refuting. And this also doesn't seem resolvable to any significant degree (it's unfalsifiable), because someone can just maintain that their own personal experience cannot be explained naturally, regardless of how much evidence is presented, and there isn't really any way to refute that, because it relies entirely on an experience you aren't privy to. You could possibly try to establish some common ground and work from there, although that likely won't be effective. As for me, my thoughts do seem to be quite bound by my desires, preferences, biases and mental limitations, as well as by my environment, none of which I choose* (and I don't think anyone would really disagree that they have similar restrictions). This aligns well with naturalism, because this is exactly what naturalism says. The only question is whether there's anything besides that influence, but vaguely claiming that there is such a thing is little more than an argument from ignorance: it seems entirely unnecessary, as it doesn't provide any explanatory or predictive power, so I'd reject it based on Occam's razor.
* I mean, I could potentially change my environment, but if my thoughts are already determined partially by my environment, that presents somewhat of a chicken-and-egg situation, where I could change my environment to change how I think, but the thought to do so is influenced by my current environment (and here we do know which one came first: it's your environment). I could potentially also reflect on my biases, work on my limitations, etc., but those involve the same problem of the thoughts to try to change them already being influenced by the things you're trying to change.
Note that some non-natural consciousness also seems to come with a massive epistemic burden, in that now you're claiming there's an entire existence beyond the natural, and this somehow interacts with the natural world, and maybe this interaction goes both ways, or maybe it doesn't, and maybe there's some universal consciousness that we all belong to or maybe there's a consciousness for each of us floating around somewhere, and that somehow attaches to each of our bodies upon birth, I guess, and maybe those stick around after we die. As purely a hypothetical philosophical idea, it might be okay, but when you try to make sense of it in the real world, given observable reality, it seems to become rather flimsy and highly doubtful.