Spoiler: It won't. (No one's quite sure how the rumor even started, which should tell you something right there.)
The price of the next iPhone — which, like the "new iPad," is probably simply going to be called the "new iPhone" — is going to be the same as the current iPhone and every iPhone before it, back to the iPhone 3G. That's how Apple works: Product prices stay the same, or they get lower, even as the products get better. Which means the entry-level version of the next iPhone will run $199 or lower on contract. Just like it does right now.
There is a way to pay $800 for an iPhone, if you really want to: buy it unlocked, off-contract. A 64GB unlocked iPhone runs $850, while a 32GB model runs $750. In case you're unaware of how phone prices and contracts work, your smartphone — whether it's an iPhone or an Android phone or a BlackBerry — actually costs hundreds of dollars. When you agree to sign a 2-year contract with your carrier, they cover most of the price, so you only pay $200 or $100 — or sometimes nothing — for the phone itself. That's because you'll be paying your carrier thousands of dollars over the course of those two years, more than making up for the money they fronted you on your new phone.
So buying an unlocked iPhone that can use on any GSM carrier is the only way you'll pay $800 for it. Oh, and the new Siri won't cook, clean, do your homework or anything remotely unseemly. But hey, you win some, you lose some.