Question: Why can one forgive a loan on Shabbos if there is a rabbinic prohibition against doing business on Shabbos?
A. The Mishna Berura (306:6:33) says that business is forbidden on shabbos whether it is just verbally or by actually handing over something, as a rabbinic ordinance lest one might come to write on Shabbos. It is also forbidden to give a gift to a friend because it is like business since there is a transfer of ownership. However the Rabbis permitted a gift to be given if it can be used on Shabbos or Yom Tov.
B. Business usually involves an acquisition in which people write up documents as proof that an acquisition was made from one person to another.
Answer: The Dirshu footnote says in the name of Rav Elyashiv that forgiveness of a loan does not need any acquisition therefore it is absolutely permitted on Shabbos. The Responsa Torah Lishma adds that forgiving a loan is unlike business since it is only a removal of a debt. It is a passive act which does not need to lead to any active actions. But one should not do it in front of witnesses because that would be under the category of uvda di’chol, doing mundane things on Shabbos.