Most modern audiences understand the cultural divide between us and Shakespeare's audience. The issues around gender were differently voiced then than now. Some of the lines in the play have been viewed as misogynistic by modern critics. Especially troublesome (for modern viewers) is Act Five, scene 2, especially Kate's closing scene where she speaks of obedience to one's lord and the importance of placing one's hand beneath one's husband's foot. A modern audience squirms at Kate's admonition, and rebels against the very notion of subservience. (Also difficult for a modern audience is the notion of men betting on the obedience of their women). The difficulty in "playing" this play to a modern audience is in the directing. How do we see Kate's delivery of these lines? Is she truly subservient? Or, is she playing. It is difficult to see Kate as someone who has lost her wit completely (although she does say she her mind and her reason were *once* big). I've seen two excellent interpretations of this scene at Vancouver's Bard on the Beach Shakespeare festival. One actress delivered the line straight, the other with a slight ironic smile. Each of these renditions offer a different interpretation of Kate's transformation. In one, it is complete. In the other, her personality and behaviour has shifted but her core wit and humour has not.
I teach this play every year, and the one major problem I run into with my students is the difference in belief systems then compared to now. Today's audiences are relatively unfamiliar with the idea of a dowry which is Petruchio's sole motivation for marriage. If students don't understand his motivation, the basis of him seeking marriage is lost.
Another concept that students have a had time grasping is Baptista's insistance that Kat find a husband before Bianca.
The last one that causes problems is the actual "taming." Some of my students don't understand why it is necessary for Pet to tame Kat in the first place. They witness marriages and couples that are just like this one or even worse. The idea of female subserviance is quite lost on them now.
Take those things out of TOS and there isn't much comedy anymore!