Generally mortgages contain provisions that if the property is sold or foreclosed upon, the outstanding loan obligations secured by the mortgage become immediately due and payable. It is important to review the terms of your mortgage to see if it contains such a provision.
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Atty Walton generally has a good answer. Rather than starting a separate interpleader action, you may simply be able to join the foreclosure action as a third party and ask the court to decide where the rent should go. This would be a lot easier and less expensive for you. Some judges will permit this, ohters will not - depending on the judge and local court rules. It is worth a try.
Your question raises several issues. Is there a deficiency judgment being sought by the mortgage holder? If not, end of story. In a short sale with the borrower leaving the state, there may not be so that the redemption period can be shortened. Check on that first. Generally, when one seeks to enforce a judgment obtained in a different state, the obligation arises out of the state where the judgment was obtained. Therefore, if the mortgage holder is seeking a deficiency judgment, you...