Readings on the gender roles and relations in The Taming of the Shrew have changed throughout the years as the social norms surrounding gender have changed. While for years audiences enjoyed the portrayal of traditional gender roles with the men dominating their subservient wives and with Petruchio taming away Katharina's "shrewishness," modern audiences are understandably uncomfortable with such an unequal portrayal of marriage. In modern productions of the play, it is more common to play Petruchio's taming as a semi-ironic act, with him and Katharina playing at and mocking the traditional roles rather than actually subscribing to them.
Though Ten Things I Hate About You does not rely on such dated and uncomfortable tropes for its gender relations, it does still have several thematic similarities to its source material. For one, women are viewed as prizes to be won in both the play and film; consider how Cameron and Joey talk about Bianca. She is clearly a status symbol as the most beautiful girl in the school. In Kat's case, Patrick will even get cash if he wins her affection. The competition to woo a woman to gain her as a prize, as well as the status that comes with her, is a central plot point and theme in both works. Additionally, by ending the story with Kat and Patrick's reconciliation (and the probability that they will become the kind of high school couple that Kat despises), the theme of changing and taming oneself for love is definitely present.
Still, while the film relies on these tropes for plot purposes, one could definitely make the argument that they are also challenged. When Bianca philosophizes about "like" and "love" with the very quotable line "I like my Sketchers, but I love my Prada backpack," she is not only demonstrating the shallowness of her character, but also making some commentary on the absurdity of teenage romances. How could people who ponder so heavily their feelings on material objects be taken seriously about their relationships with other humans?
Another point in the film's favor is that Cameron is portrayed as more "pure at heart" than Joey, who clearly is only looking for sex with Bianca. The audience roots for Cameron and it is definitely a satisfying moment for feminism when Bianca defends her own honor and Cameron's by punching Joey in the nose after he knocks Cameron down. This flipping of gender norms definitely challenges traditional roles.
Finally, the reconciliation of Kat and Patrick can also be understood as challenging the norms and expectations society has for them. After all, even though their rough edges have been smoothed out, it seems to have been a mutual decision to do so. It's also important to note that Patrick has changed and opened himself up significantly as well, telling Kat where he really was during his secret summer and letting his "tough guy" act drop in front of her. This mutual letting down of barriers is far from the taming of Katharina in the play, which is clearly one-sided. Overall, I would argue that the film challenges the gender roles of The Taming of the Shrew.