I just moved to MN from PA and have a place to live here, pay bills here, have health insurance here, and intend to stay. All these things mean I will file my 2020 taxes next year as a MN resident. But I want to vote in the primaries. MN's primaries are over but PA's isn't til June. If I vote (absentee) in PA's primary in June and don't change my voter registration to MN til afterwards, will that mess up my 2020 residency? I don't want to do PA taxes next year.
For tax purposes, your domicile is
"WHERE AM I DOMICILED? A domicile is the place at which an individual maintains his/ her permanent abode and to which he/she intends to return whenever absent. A person may only have one domicile at any one time."
See this Pennsylvania webpage.
If you look at that link at the number of factors that go into determining your domicile, it seems to me that you have already relocated your domicile to Minnesota.
Whether or not you have to file a PA tax return depends on - among other things - when you moved from PA to MN. I can't tell from your question if you moved before or after January 1st.
As for the legality of you voting in the PA primary after having moved to another state, that is properly a question for a voter official in your home county in PA, or, I suppose, the Secretary of State (if that official is the senior elections official in the state). We at TurboTax are not qualified to answer for that.