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!