As noted herehere, R. Yosef Albo (Sefer ha-Ikkarim 3:1) writes that:
Coral is intermediate between inanimate matter and plants. We also find the sea sponge, which only has the sense of touch, and is an intermediate between plant and animal stages. We also find the monkey to be intermediate between animals and man.
This idea is repeated, for example by R. Moshe Isserles (the Rema) in his commentary מחיר יין, on מגילת אסתר, א:ו, and is ultimately based on Aristotle's classifications. This idea is also cited in the name of the Arizal (see, e.g., Ne'ot Ha-Deshe, Devarim 2). However, this was not written in a halakhic context, and it is unclear whether it would have halakhic ramifications.
In a halakhic context, the Torat Kohanim (Parshat Shemini, 6:5) explains the verse כל הולך על כפיו to include the monkey in the law of נבלת בהמה. The Ravad in his commentary explains:
והא דאמרינן הולך על כפיו זה הקוף רבותא קאמר שאעפ"י שדומה בצורתו לאדם ובאצבעות ידיו ורגליו כאדם אעפ"כ טומאתו כבהמה
In other words, the Torat Kohanim requires an additional דרשה to teach us that monkeys are included in this law, because otherwise we may have thought that monkeys follow the rules for humans with regard to טומאה.
See also R. Chaim Kanievsky's discussion of whether one can write a tattoo on the species known as אדני השדה here. He concludes that there may be a three-way dispute as to whether it is viewed as a plant, animal, or human.