Does the renter have to pay rent to the owner if it is in foreclosure in Minnesota? And if the lease that's signed continues after foreclosure does the renter have any rights to stay in house?

Just about everywhere in the U.S. the folllowing answers will apply:

The renter owes the rent to the current owner until the foreclosure sale happens.
The renter can stay for at least 90 days after receipt of a notice to quit from the new owner, longer if renter has a lease. It's Federal law. See this (https://www.askmehelpdesk.com/real-estate-law/paying-rent-when-property-foreclosure-316199-2.html).

What if the sale happened but is not final and no foreclosure papers have been posted?

Although an owner (whether it's the lender who foreclosed, or a new buyer) is likely to be very attentive to making sure they collect your rent rather than the old owner collecting it, it doesn't hurt to find out what is going on. It sounds like you have to continue to pay the old owner. But you can't be forced to pay twice as long as you have canceled checks proving you paid, so try not to worry.
A sale can't 'happen' until it's final. Many a closing falls through right at the closing. It isn't final until everyone walks out of the room with money and papers traded hands.