Generally speaking, living abroad temporarily (where the word "temporary" here can mean a period from as short as a few days to as long as an indefinite number of years) does not change someone's legal domicile (a well-established legal concept). Thus, moving abroad doesn't normally change a United States taxpayer's responsibility to file and pay taxes to their "home" state.
However, New York is something of an exception to these otherwise normal rules. Please allow us to explain this.
Typically, New York taxpayers living and working in another state, or in a foreign country, have to pay taxes to New York state on all of their earnings, wherever in the world they earn income. Moreover, they do not change their full-year New York resident status by simply living in another state or country -- as long as they maintain their New York domicile.
However, New York does allow some exceptions to this, in contrast to the tax practices of most other states, for those New York residents who live overseas for an extended period of time.
It seems that you could very well qualify for such an exception, if you've been living in Chile "long enough." In order to determine that with certainty, however, and to help you define your own residency status for New York state tax purposes, please refer to the following New York Department of Taxation & Finance document (you can click on the link to open it):
https://tax.ny.gov/e-services/training/preparer/pdf/Residency%20and%20Allocation%20Rules%20for%20NYS...
In particular, please see Pages 6 through 10 (Slides #'s 5 - 9).
See also Pages 13 through 23 (Slides #'s 11 - 21), and the discussion there on "Nonresident Domiciliaries." As missionaries living in Chile, it appears possible that you may qualify for this tax benefit. New York state nonresident tax status, for income tax reporting purposes, is granted to those otherwise New York residents living overseas, even if they maintain their permanent legal domicile in New York and will eventually return to it, if they can satisfy the tests described therein.
If you can meet those tests, by virtue of living in Chile, then you can qualify as New York nonresidents for state tax purposes (although you still are required to file a US federal tax return). In such an event, you may not even have to file a New York tax return if you don't want to. For more information on the requirement to file a New York tax return while classified as a non-resident, please see the following webpage:
https://www.tax.ny.gov/pit/file/nonresidents.htm
Thank you for asking this important question, and good luck with your missionary work.
