My employer is located in MA.

Next year I have the option to work on average 3-4 days from home in NH and 1-2 days in an office in MA.

Will I have to pay MA state tax?

I spend 30% of the year outside of MA and NH (various US, London and Rome - no location more than a week).

Does this affect my residency for tax purposes?

Probably. Massachusetts will considre you a full-time MA worker if you consistently work in the state.