If your resident state collects income tax, you're still required to pay taxes to your resident state on any income that was earned in a "tax-free" state.
Example: You live in Oklahoma and work in Texas. Oklahoma collects state income tax, but Texas doesn’t collect state income tax. Even though you work in Texas, your income will still be taxed in Oklahoma.
But what if you were a Texas resident working in Oklahoma? You'd still have to file a nonresident Oklahoma return. Being a resident of a "tax-free" state doesn't exempt you from paying state taxes on income earned outside the state.
Now, if you were a Texas resident working in Alaska, you wouldn't have to worry about any of this, as neither state collects income tax.