While it may technically be legal and appropriate for a US Permanent Resident living in Canada to file a US non-resident tax return (1040NR), as an immigration lawyer with over two decades of experience helping Canadians maintain their permanent resident status, I can tell you that it won’t be looked on favorably by either US Citizenship and Immigration Services in adjudicating naturalization petitions, or U.S. Customs and Border Protection when one is attempting to cross the border into the U.S.
Border officers won’t actually have that information in their database when a green card holder tries to cross, but if that individual has been resident in Canada for any length of time, they will almost certainly ask how the green card holder’s taxes have been filed. If the answer is as a “non-resident,” you can bet they’ll be shown into secondary inspection and questioned extensively about the ties they have maintained to the United States and other indicators of what their intention was when they left the U.S.
Border officers tend to be very aggressive about trying to get nonresident green card holders to surrender their green cards, and this gives them extra ammunition. Filing a non-resident return is an indicator that the individual lacks the intention to continue to be a resident of the U.S., which is the legal standard that will be considered by an immigration judge should the matter ever be adjudicated.