In a series full of ruthless trope subversions, Tyrion-as-hero (or at least as sympathetic underdog) is a plot that's easy to get behind. He's highly intelligent and well-read, he speaks bluntly about how things "really are," he's constantly making sarcastic comments, and, unlike most powerful people we see in the series, he wants to do right by the people of Westeros. Throw in the fact that his efforts and intelligence go unrecognized, due to the fact that he's ugly and a dwarf, and he makes the perfect reader stand-in as the unappreciated (but highly deserving) hero.
But Tyrion's plotline is not only a subversion of "the handsome man is the hero, the ugly man is the villain." It's also a subversion of the entire concept that the underdog is the true hero. Although Tyrion is an interesting character, he is not always an admirable one, whatever he might believe. He deals with his own feelings of powerlessness by asserting his power and his superiority over others who are even more powerless, aka women. He is, despite his own feelings of benevolence, deeply misogynistic.
And, as compelling as Tyrion's storyline may be, readers' eagerness to defend him is more than a little uncomfortable.
This post contains spoilers through A Dance with Dragons.
Tyrion makes little comments and observations throughout the books that hint at his misogyny, but the first time it became clear, to me, was through his relationship with Shae. It is difficult to figure out the truth of their relationship, as we are stuck in Tyrion's head and Shae is quite enigmatic, but Shae's frequent references to jewels and money, and Tyrion's initial insistence to himself that she's a "whore" who doesn't truly care for him, strongly suggest that Shae is pretending to love him for her own security and advantage. She is clearly a survivor, a savvy young woman who has lived through a lot, and The Hand of the King provides the best opportunity for her in a land that is being ripped apart by war.
And she provides an excellent opportunity for Tyrion as well. After his experience with Tysha, Tyrion has convinced himself that no one other than a whore would ever want to be with him -- and this suits him, as any relationship with a prostitute puts him in the powerful position he craves. Shae, as the beautiful, strong young woman who acts like she loves him, is the perfect person for him to fall in love with (or at least, believe that he loves). He is always in a position of benevolent power over her, providing her with money and jewels, manipulating her into a position as Sansa's maid, and acting as her protector against forces that only threaten her because of his own interest.
This same sense of benevolent power pervades his marriage with Sansa. He is appalled at the idea of marrying her, as she is "no more than a child," but when the wedding night comes, he is highly attracted to this innocent, pure, beautiful, terrified, defiant girl who is completely under his power. He will not sleep with her -- not until she wants him to -- but there's a sense that this basic decency also gives him great pleasure, as it bolsters his opinion of himself as the heroic figure. He then becomes frustrated that she does not recognize the benevolent part of his power or grow to trust him. The same defenses that he admired in her before their marriage become frustrating to him, because she should recognize that he is different. He is good.
Of course, there are many male characters who display far worse misogyny than this strange virgin/whore dichotomy, benevolent power complex that Tyrion expresses. But Tyrion's luck changes drastically at the end of A Storm of Swords, when the idea of him as the underdog is ripped to shreds by a family that will never appreciate him and by his own response to these events.
He strangles Shae for "betraying" him in court, but it is hard to know what was really going on. Was she acting when she delighted in embarrassing him before the crowd, in order to save herself from Cersei? Or was she acting when she cried in front of him and begged him to save her? In the end, it doesn't really matter. Tyrion kills her for betraying him... but he also kills her for failing to live up to the woman he had created in his head, the prostitute who loves him exactly as he hopes. For failing to appreciate the benevolent way he treated her, and all the advantage he brought. As he strangles her, using the chain of the Hand of the King, the benevolent power that protected her turns into a death sentence. Because he is only benevolent when others support his self-image as the unappreciated hero. And throughout A Dance with Dragons, he never expresses regret or shame for what he did (except, perhaps, regret that she made him do it).
At the start of A Dance with Dragons, Tyrion is a broken man -- betrayed by Shae, betrayed by his brother, set up to die by his family, having lost everything -- and he channels that into asserting his own, not-so-benevolent power, making others feel even more powerless than himself. He thinks of all women as "whores," and is determined to evoke fear in the first one he meets, threatening to strangle her not because he intends to, but because he wants her to recognize, and fear, that he is the one with the power here. He is certainly not a noble character here. Tyrion has good traits, but his experiences have forced them out of him, leaving him as twisted as everyone expects him to be.
Finally, there's his planned punishment of Cersei: he intends to rape and kill her when he returns to King's Landing. From the perspective of an underdog, Cersei deserves punishment. She attempted to kill him multiple times, she set him up, and she betrayed him. But perhaps worst of all to Tyrion, she outplayed him, beating him at his own game of intelligence and manipulation. He wants to both punish her and reassert his authority over her before killing her, and the best way to do that, it seems, is to punish her for being a woman, by reminding her that, as a man, he still has one major power advantage over her.
Tyrion shifts from a subtle, benevolent kind of misogyny to a twisted, overt sort in one moment, but readers are often still eager to excuse him and his behavior. "He had to kill Shae. She betrayed him!" "His threat to rape Cersei wasn't misogynistic; he only said that because she is evil." The TV show, by turning him into the noble protagonist and making his relationship with Shae into a true love story, seems to be following the same path. And it is easy to want Tyrion to be the hero. We want him to be the underdog and represent good in the series, and the fact that the underdog, as an ugly and unappreciated figure, feels like a trope subversion lulls us into believing that this narrative is safe from George RR Martin's ruthless plots twists. And once we identify with his character, it is hard to let go. Yet the underdog and the unappreciated hero are both also tropes, and George RR Martin twists it again, so that the character who is ugly on the outside actually is sometimes ugly on the inside. Because if everyone treats you as though you are wicked and broken, you might well end up fitting their beliefs. Tyrion is a fascinating, frequently enjoyable character. But he also is frequently repulsive, not in appearance, but in his thoughts and actions. And that, in some ways, makes him more interesting. But it does not make him a hero.