No, when you e-file you do not need to send your out-of-state return to MN when claiming the credit.
However, the necessity of the M1CR credit itself depends on the type of income you earned. Minnesota and North Dakota have a tax reciprocity agreement. If your only income from North Dakota consisted of wages, salaries, or tips, you typically should not be claiming a credit for taxes paid to another state. Instead, under reciprocity, that income is only taxable in your home state of Minnesota. If North Dakota taxes were withheld from your paycheck in error, you should file a North Dakota return to claim a refund of those withholdings rather than claiming a credit on your Minnesota M1 form.
Just confirm you are using Schedule M1CR to claim the credit if you have income that falls outside of the reciprocity agreement, such as income from a North Dakota farm, rental property, or a business partnership. If you fall into this category and choose to file your Minnesota return by mail rather than electronically, then you must include a copy of your North Dakota return along with your Minnesota forms .
Even when e-filing, keep a copy of your signed North Dakota return and all supporting schedules in your records. The Minnesota Department of Revenue sometimes requests these documents later to verify the amount of credit claimed.
Here is more information on reciprocal states: What is a state reciprocal agreement? and here is a TurboTax article you may find helpful: How do I file a nonresident state return?
