If somebody is owed money, is it permissible to formally waive the debt on Shabbat, or does this violate the prohibition against legal transactions on Shabbat?

There is a debate among the authorities as to whether waiving a debt requires a Kinyan – a symbolic act of acquisition. If a Kinyan must be performed to formally waive a debt, then this would, indeed, be forbidden on Shabbat.

Chacham Ovadia Yosef, in his work Halichot Olam (vol. 3, p. 170), rules that Halacha distinguishes in this regard between loans that are taken with a Shetar (legal document), and those which involve a purely verbal commitment. If a Shetar was written, then waiving the debt requires a Kinyan, and is thus forbidden on Shabbat. In cases, however, of a purely verbal agreement, no Kinyan is necessary, and the lender would thus be allowed to announce his waiving of the debt. All the more so, Chacham Ovadia adds, in a case where the borrower is poor, such that waiving the debt constitutes a Mitzva, it would be permissible for the lender to cancel the debt on Shabbat.

Summary: A person who is owed money may waive the debt on Shabbat if there is only a verbal agreement between the lender and borrower; if they wrote a contract, then the lender may not waive the debt on Shabbat.