You sure do.
When you buy the iPhone, all plans have an unlimited internet package included, which is why the package with the least amount of minutes costs you $69.99.
If you have a family plan and want an iPhone, they'll charge you $29.99 for the unlimited data plan (that's on top of the $9.99 add-a-line and the family's minute plan).
AT&T will not sell an iPhone without including the unlimited data (internet) plan, so if you want the phone, you must pay for internet access, even if you won't use it (but if you want an iPhone, why wouldn't you?)