Should the amount listed in box 14a of a K1 be exempt from I&D taxes in New Hampshire? How can you select a reason code when that info isn't in the k1?

Is the K-1 from a partnership, S-Corp or Estate/Trust?

The K1 is from a trust in Oklahoma.  I'm a beneficiary but resident of NH.

No, the amount listed in Box 14A of the K-1 (Trust) should be taxable in New Hampshire.  This entry is for "tax exempt interest income,"  which is not taxed on the federal return.  Since it comes from Oklahoma, it is taxed in New Hampshire.

To reflect this on your return, please follow these steps:

1. Go back into your federal return, to the K-1 section. 
2. On the Tell Us About Your Schedules K-1 screen, click the Update box next to Estates or trusts (Form 1041).
   
3. On the Estates and Trusts K-1 Summary page, click Edit next to the K-1 you entered.
4. Continue through the K-1 interview, on the Enter Box 14 Info screen, select code A from the list and enter the amount on the K-1.
5. On the screen Enter Income Detail, select Oklahoma from the list and enter the amount.

This will ensure the figure flows properly to the NH DP-10.

If I do this, will I need to file an OK tax return even if I have no taxable income in that state? (I'm a non-resident.)

Maybe, depending on the amount of income from the K-1.  OK Tax Commission states: "Every nonresident with gross income from Oklahoma sources of $1,000 or more is required to file an Oklahoma income tax return. Use Form 511NR."