In California
My lender is in the process of resetting my mortgage. Can my month-to-month renter refuse to pay rent?

Also, would the lender to to the property address and tell the renter that the we were in default when they knew all along we lived at another address a few doors down the street?

Often a John Doe notice is sent to the address informing them of the default.

Unless you live in California, they have to pay the rent. If they don't you can evict them. In California, they do not have to pay rent and can even sue for their rent back for the time you were in default.

California has a law on the books that requires a landlord to apply rental payments fully towards the mortgage of the rental property. If they don't and the loan goes into default, the tenant can stop rental payments and sue. This law is also limited to landlords in the first few years of ownership.

Also it is normal for a lender to post notice on the property that that a foreclosure is being processed.

So, if you come under the provisions of that law, the tenant can refuse to pay. Otherwise they have to continue paying or be subject to eviction.