I see it cited all the time by the fandom that Hermione is "close-minded" - citing Xenophilius Lovegood, Luna's father, from Deathly Hallows.

[…] “But then…do you mean…” said Hermione slowly, and Harry could tell she was trying to keep any trace of skepticism out of her voice, “that you believe these objects - these [Deathly] Hallows - to actually exist?”

“Well, of course,” [said Mr. Lovegood.]

“But,” said Hermione, and Harry could hear [Hermione’s] restaint starting to crack, “Mr. Lovegood, how can you possibly believe -?”

“Luna has told me all about you, young lady,” said [Mr. Lovegood]. “You are, as I gather, not unintelligent, but painfully limited. Narrow. Close-minded.”

[…] “All right,” said Hermione, disconcerted. “Say the Cloak existed…what about the [Resurrection] Stone, Mr. Lovegood?”

“What of it?”

“Well, how can that be real?”

“Prove that it is not,” said [Mr. Lovegood].

Hermione looked outraged.

“But that’s - I’m sorry, but that’s completely ridiculous! How can I possibly prove it doesn’t exist? Do you expect me to get a hold of - of all the pebbles in the world and test them? I mean, you can claim anything’s real, if the only basis for believing in it is that nobody proved it doesn’t exist!”

“Yes, you could,” said [Mr. Lovegood]. “I am glad to see you are opening your mind a little.”

[…] Harry glanced at Hermione. She was frowning at [Mr. Lovegood], but she did not contradict him.

[…] “What do you think?” [Harry] asked Hermione.

“Oh, Harry,” [Hermione] said wearily, “[His story] is a pile of utter rubbish. This can’t be what the sign really means. This must just be his weird take on it. What a waste of time.”

However, the thing is, even though the Hallows were, indeed, real - Hermione wasn't wrong about Mr. Lovegood's argument being a "pile of utter rubbish".

Why? Because Mr. Lovegood literally used a fallacy to try and make his point.

When he says:

“Prove that [the Resurrection Stone] is not [real],” said [Mr. Lovegood].

He is committing the burden of proof fallacy - which is why Hermione looks "outraged". This is also bad reasoning that is widely abused in arguments.

According to Rational Wiki:

Fallacious shifting of the burden of proof occurs if someone makes a claim that needs justification, then demands that the opponent justify the opposite of the claim (i.e. "Prove that it is not real!").

The opponent has no such burden until evidence is presented for the [original] claim.

Often, someone will present a new idea and say that it must be accepted because it cannot be disproved. This is insufficient because without evidence there is no reason to accept an idea, even if there is no contrary evidence.

One example is that of a simulated reality, which proposes that the human race does indeed live in The Matrix, and we are a computer simulation. There is no evidence against this idea, in fact, it may be impossible to fully disprove, but as there is no real evidence for it there is no reason to accept the idea as real. Another famous example is the teapot proposed by Bertrand Russell ("Russell's Teapot"), the existence of which cannot be disproved.

Indeed, if one is making an exceptionally bold claim (such as with alternative medicine that claims miraculous cures) then exceptional evidence is expected in its support.

If we are to consider the concept of Russell's Teapot:

"If I were to suggest that between the Earth and Mars there is a china teapot revolving about the sun in an elliptical orbit, nobody would be able to disprove my assertion provided I were careful to add that the teapot is too small to be revealed even by our most powerful telescopes.

But if I were to go on to say that, since my assertion cannot be disproved, it is an intolerable presumption on the part of human reason to doubt it, I should rightly be thought to be talking nonsense.

If, however, the existence of such a teapot were affirmed in ancient books, taught as the sacred truth every Sunday, and instilled into the minds of children at school, hesitation to believe in its existence would become a mark of eccentricity and entitle the doubter to the attentions of the psychiatrist in an enlightened age or of the Inquisitor in an earlier time."

Note the points in Russell's original argument that I bolded above. Both also fit this scene:

• Mr. Lovegood implies that Hermione's logical reasoning is "intolerable", calling her "close-minded" for simply calling into question his belief (and rightly so) that the Deathly Hallows, indeed, exist.

• Consider the tale of the Three Brothers/ Deathly Hallows - they are literally taught, through The Tales of Beedle the Bard, to almost all wizarding children, especially in Pureblood or ancient magical families. However, Hermione is a Muggle-born, and was raised by Muggle dentists - she never grew up with these stories. Therefore, her hesitation to believe in the tale of the Hallows - just like someone using "The Little Mermaid" as justification to believe in the real existence of mermaids - is seen as "eccentric" by Mr. Lovegood.

Still, Hermione is absolutely right in asking Mr. Lovegood why she, or any of them, should believe his claims - and she has good reasons why they shouldn't. After all, Mr. Lovegood is expressly wrong, or in denial, about other things, even in the face of facts and evidence.

From just a short time before the "close-minded" statement:

[…] Hermione let out a small cry of shock.

“Mr. Lovegood - what’s that?”

She was pointing at an enourmous, gray, spiral horn, not unlike that of a unicorn, which had been mounted on the wall, protruding several feet into the room.

“It is the horn of a Crumple-Horned Snorckack,” said [Mr. Lovegood].

“No, it isn’t!” said Hermione.

“Hermione,” muttered Harry, embarrassed, “now’s not the moment -”

“But Harry, it’s an Erumpent horn! It’s a class B Tradeable Material, and it’s an extraordinarily dangerous thing to have in a house!”

“How d’you know it’s an Erumpent horn?” asked Ron…

“There’s a description in Fantastic Beasts and Where to Find Them! Mr. Lovegood, you need to get rid of it straightaway, don’t you know it can explode at the slightest touch?”

“The Crumple-Horned Snorckack,” said [Mr. Lovegood], very clearly, a mulish look upon his face, “is a shy and highly magical creature, and its horn -”

“Mr. Lovegood, I recognize the grooved markings around the base, that’s an Erumpent horn, and it’s incredibly dangerous - I don’t know where you got it-”

[…] “…I agree with Ron,” said Hermione. “[Mr. Lovegood is] an awful, old hypocrite, telling everyone else to help you, or trying to worm out of it himself. And, for heaven’s sake, keep away from that horn!”

Again, Hermione's not wrong - the "stubborn, mulish" Mr. Lovegood refuses to be anything but "close-minded" when Hermione challenges his view that the horn is that of a 'Crumple-Horned Snorckack'.

However, when Hermione again confronts Mr. Lovegood about his views, instead of providing evidence, he criticizes her - calling her "close-minded" - instead of refuting her reasoning.

Here, we see Mr. Lovegood willfully endanger himself, his guests, and even his own daughter by keeping an extremely dangerous - and very illegal - object in his own home...and, thus, Hermione has good reason to believe he's in willful denial about the Hallows.

Yet somehow - and still - it's commonly cited in the Harry Potter fandom, as if it's fact, that Hermione is "close-minded" (as Mr. Lovegood claims), despite it being largely a misconception.

Hermione even - again, correctly - points out all of the flaws with Mr. Lovegood's reasoning. It's because of this flawed reasoning that Hermione is disbelieving - not because she's stubbornly against considering that the Hallows may actually exist, but because she has no logical reasons (or evidence) to believe so.

As to why it is important to avoid fallacies:

[...] ...we attempt to avoid fallacies because we care about what is true, and we want to believe what is true, and not what is false (at least, when we are being reasonable). So we want to avoid reasoning that does not help us (and may actually hinder us) from our pursuit of truth. (Source)

The inability to think critically makes us vulnerable to manipulation, and opens us up to making bad decisions based in bad reasoning. (Source)

And, again, true to form, Hermione was correct in not accepting Mr. Lovegood's reasoning and argument - because, at the time, he was trying to manipulate them into making bad decisions (i.e. staying long enough to be captured by Death Eaters).

Hermione was rightfully suspicious, having also noted Mr. Lovegood's "red flag" behavior - so why, then, do fans give her so much flak for doing what other, perceptive people would reasonably also do in her shoes?