Where an organization is located, whether is for profit or not for profit, does not require a banking relationship in the same state. With the internet and online banking, allows consumers the option to find the right financial institution whether it is across the street, or across the country.
Some of the logistics, may make it easier to use a local bank (making deposits and having access to the funds quicker), but it is not required.
The only limitation that could possibly come into pay is within the banking laws, rather than tax laws. A bank may require that a non-profit provide charter documents from within the state. I know that many non-profits open accounts in other states, whether it is bank account or a brokerage account.
Where your business is should have no effect on an unrelated 501(c)(3) organization, even if you are an officer or board member of the group.