If I find an animal trapped under a box on shabbat, is there anything wrong with letting it out?
Though I have been unable to locate a good copy of the Mishna Berura (316-25) that he quotes, this document does say that there is no issue with freeing a trapped animal.
This is also asserted by the book "39 Melochos". (There's a footnote there, but Google Books won't let me see it.)
This is also stated by "The Concise Code of Jewish Law" with no source. He notes that one should not touch the animal, as it might be muktzeh.
Chabad.org, in a footnote to the Rambam's Hilchos Shabbos, brings a Magen Avraham that says (and a Maggid Mishnah that implies) that freeing an animal on Shabbos is completely permitted provided one does not touch it, since it is muktzeh.
The Shulchan Aruch HaRav 316:12 (quoting the above Magid Mishnah and Magen Avraham) states:
מותר לפתוח הבית בפני הצבי כדי שיברח ויצא וכן צבי שנכנס למצודה מותר לפרקו ממצודתו ובלבד שלא יטלטלנו