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!
According to [the Illinois Department of Revenue](https://www2.illinois.gov/rev/questionsandanswers/Pages/10.aspx), you must file an Illinois income tax return if “you earned enough taxable income from Illinois sources to have a tax liability[.]” There is a form for residents of Iowa, Kentucky, Michigan or Wisconsin to fill out exempting them from withholding, but not Indiana. “If you received wages, salaries, tips, and commissions from Illinois employers, you are not required to pay Illinois Income Tax on this income. This is based on reciprocal agreements between Illinois and these states.”

So this might be useful if you earn a lot of passive income from investments out-of-state (although in that case the way to make them tax-free *legally* is to buy muni bonds), or if you want to work in Indiana but live in Chicago, but not if you work and live in Illinois.

Most people who work in Illinois would have state income tax withheld, although the hypothetical tax cheat would have several ways around that.