I would argue that the narrator's unreliability becomes obvious when she starts imagining the wallpaper to be observant and malicious. The fact that she starts to see a figure in the wallpaper's design makes it abundantly clear that she is not of sound mind and therefore not reliable.

Her constant tears and feelings of sadness and weakness add to the idea that she is suffering from some kind of mental health disorder. She is spending her time watching her wallpaper, imagining it to be a dynamic entity of some kind.

As the story progresses, it becomes increasingly apparent that the narrator's perspective is not to be trusted. She refers to a "yellow smell" emanating from the wallpaper and filling the house and the image of woman attempting to climb through the wallpaper. She also has a mistaken impression that her husband, John, and his sister, Jennie, are also trying to solve the wallpaper's mystery.

The final journal entry displays a complete breakdown in the narrator's mental health, and by this time, it is impossible to think of her as a reliable narrator. She eventually strips the wallpaper with her bare hands in her attempt to free the trapped woman. These are not the actions of someone whose words are to be believed.

The narrator's evolving relationship with the wallpaper in her bedroom demonstrates her mental breakdown, and her inability to recognize her deteriorating mental health helps readers to recognize her as fundamentally unreliable. To be clear, she does provide a reliable narration of her own experiences as she experiences them, but we cannot rely on her to provide objective or accurate information about much else.

At first, she simply hates the wallpaper, and she feels that it could be "one reason [she does] not get well faster." She analyzes the paper, saying that it commits "every artistic sin" and that its lines "commit suicide": extreme and troubling remarks, but they also provide evidence of her education and intellect and imagination. Soon, however, she says,

This paper looks to me as if it knew what a vicious influence it had!

Now she is ascribing consciousness and intention to the wallpaper, suggesting that it is actually aware of its effect on her and is, perhaps, purposely attempting to produce this effect. It does not take long for her to decide that she might be improving "because of the wallpaper." Without stimulation, and absolutely beset with boredom, the narrator has nothing to think about but the wallpaper, and she begins to feel that there are two patterns in the paper. She admits to lying awake "for hours trying to decide whether that front pattern and the back pattern really did move together or separately." She comes to see life as "very much more exciting now that it used to be" because she has "something more to expect, to look forward to, to watch": she means the wallpaper.

It is not a sign of good mental health to look forward to one's life so that one can study the life of one's wallpaper. The narrator does not realize that this is not really a positive mental development. She's becoming obsessed with the paper, with keeping her sister-in-law from understanding it; she even grows possessive of it. The narrator develops the idea that there is a woman behind the first pattern of the wallpaper and that "[this woman] just takes hold of the bars [in the front pattern] and shakes them hard." Obviously, we know that the wallpaper is not alive, that there is no woman trapped within it, and we can understand that the narrator's descriptions of this inanimate object signal her declining mental state, though she does not realize this. This helps us to realize that she is no longer reliable as a narrator.

I think that anytime a first person narrator is used there is an element of unreliability.  As mentioned above, the narrator may certainly be credible; however, everyone perceives objective truths differently.  Even if the narrator in "The Yellow Wallpaper" is correctly relaying events as she knows them, how can we be certain that she is relaying all that has happened?  What is left unsaid?

The term "unreliable narrator" refers to a narrator who is disturbed or too sujective to be completely credible.  This narrator's recount of events is so distorted that it departs from the true understanding of these events shared between the reader and the implied author.  However, the term does not necessarily mean that the narrator is a liar or morally untrustworthy.

So, while Gilman's narrator is unreliable since she is obviously mentally disturbed, her narration presents an accurate report of her breakdown as the reader witnesses her deterioration in the story. Also, her recounts of what her husband and her sister-in-law say and do at the beginning of the story when she is fairly well, lend credibility to her reports of their treatment later in the story.