You've gotten some bad advice. Let's break this down into parts.
Working out of state. You must file a state income tax return in any state where you physically live or physically perform work, regardless of where your home state is. You file a non-resident tax return in any state where you earn income, that reports only income sourced to that state. Then, you file a resident tax return in your home state that reports all your world-wide income and pays tax on all your income, and you can claim a credit or deduction in your home state for out of state taxes you paid.
For example, suppose you live in PA, you worked 6 months in New York and 6 months in Ohio, and you have investment income besides. You file an Ohio non-resident return to report your Ohio wages, a NY non-resident return to report your NY wages, and a PA return to report all your income including your investment income, and you claim a credit on your PA return for the taxes you paid in NY and Ohio.
In your case, if your home state doesn't charge income tax, then you only file a non-resident return for any state where you earned income in that state. But you are never tax-exempt in the state where you actually performed the work, even if you are traveling away from home in order to work.
Travel Reimbursement. If you are a W-2 employee, your employer may pay you a tax-free allowance for food and lodging if you are traveling away from your tax home on temporary assignment. If they pay a stipend, they must use an accountable plan. That means that they either collect your receipts and verify your expenses, or they can use the approved federal meals and lodging rates (but they must still verify your dates of employment.) If they don't use an accountable plan, they must include the stipend in your W-2 taxable wages, and you can't deduct anything because that deduction was eliminated for W-2 employees by the TCJA.
A temporary assignment is one that is expected to last, and actually does last, less than 1 year. If the assignment lasts more than one year or is open-ended (no specific ending date) then it is not considered temporary, and your travel reimbursement is not tax-free.
