Question: Lets say you work in Chicago and live in Chicago but put your home address as somewhere in Indiana. Then every week you collect the mail that is being sent to you in Indiana. Wouldn't you save a lot of money assuming the taxes are lower? What prevents people from doing this?

Answer: 
Why stop at Indiana? Why not say you live in a tax haven in a completely different country, and not pay any taxes at all, even though you still remain in the same place?

Oh, it's called tax evasion. Right. That thing they talk about on TV all the time these days.

Many people actually do it (though it's usually a little bit more complex, they set up companies in those countries, and get the money through those companies). In some cases, it's actually even legal (see big corporations "parking" their international profits in tax havens to avoid taxation in the US), up to a certain point, though many countries are actively fighting those "loopholes".

But in many cases, it's actually just plain illegal. The country/state where you live in has laws that says you owe taxes on your income based on a number of conditions, such as your place of residence, the source of your income, and so on. In some cases, several jurisdictions may even have a valid claim to tax you (suppose you are an United States citizen living in France and working in Switzerland: all three have a reason to tax you).

There are then treaties to decide who you actually pay taxes to in those situations, but it can quickly become very complex.

If you just plain flat out lie about where you live, you'll end up being caught when they check (and they do check). You'll be asked to prove that you actually live where you claim you do. Rent, utility bills, grocery bills... If you can't prove it, you'll be taxed, and will probably have to pay penalties. And good luck reclaiming what you paid to the other state!
**Because there are much easier ways to cheat on your taxes**

While moral obligations and legal threats play a part, I believe the main reason is this would be an extremely expensive, difficult way to cheat your taxes.

**Why (Low/Middle Income)**

1) You must establish residency in a state you don't live in. A P.O. box won't cut it. You've got to show a lease or utility bills. That involves buying or leasing an actual place to live, which will be several hundred a month.

2) Per above, a low/middle net worth individual would spend more faking a residence than they'd save. It's all risk and no reward.

3) Furthermore, low/middle net worth people likely work a 9-5 job, where taxes are taken out of their paychecks before they even see the money. Claiming they don't owe state taxes is likely going to raise a red flag.

**Why (High Income)**

For high net worth individuals, it's a bit different. They can afford a lake house in a 0% income tax state, and may actually visit a weekend or two a month to get the mail and enjoy the view. 

1) They can also afford a great accountant to figure out how to legally not pay taxes in the state they live in anyway.

2) High net worth individuals are likely already itemizing deductions, and figuring out how to make the mortgage interest deduction work in their favor. They've likely also set up companies that hold money and assets for them in a tax-advantaged way.

3) High net worth individuals (many retirees are in this camp) do move to states like Florida which have no income tax. If you're running a company that does business in another state, you may still be required to pay some type of corporate income tax based on where the money was earned.