Suppose I am living in an Airbnb in NYC for a year but my permanent
address is listed as being in another state. In TurboTax, is it okay
to submit a change of address form since the federal tax return still
lists the other state as my permanent address?
Turbo tax doesn't care where you want your mail sent. Somebody could move in early 2023 so they want any correspondence sent to their new place, but would have zero impact on your 2022 taxes.
The numbers on your Federal return only depend indirectly on where you are living.
The indirect part is because where you live and work determines your state and local taxes, which if you itemize can make a difference. Normally your return reflects what happened last year but being a resident of a state may impact your ability to claim a deduction or credit at the state level. But this would be an edge case if the impact was reflected in the return you are filing in April 2023.
Note that when I fill out the taxes for the permanent address in the
other state, I have to pay a lot. When I fill it out for the address
in NYC instead, I don't have to pay anything.
Note: the other state is Pennsylvania.
New York is very aggressive in claiming income. If you were working in New York, they will want their money.
If you were living in New York, they will want your money.
You will have to determine when or if Pennsylvania ever considered you a resident of New York. Or if they think of you as a resident of PA, who is temporarily in New York.
If you had tax withholding for Pennsylvania, you will have to file a state tax return with them to get some or all of it back. If the withholding was for NY, then you may still have to file a return with PA, if they consider you an resident at amy point in 2022.
Expect to have to file 2 state and 1 federal return.