Short answer is: no. (Though can only speak regarding my country's case law) I am saying this based on the lack of such supreme court decisions in Turkey.
As long as you say "including" it will merely mean that the list is not exhaustive as another answer stated. the "but not limited to" will not change the Court's perspective on whether the list is indeed interpreted to be exhaustive or not. You can find more information on this page and the links given albeit from 2015: About "Including but not limited to" by Ken Adams
Using your example, even if the second half was not there, the phrase "The business isn't responsible for injuries from animal bites." would be valid and in court, the judge would interpret the case in front of him to be an animal or not. However, if somebody gets bitten by a creature that could be considered as an insect or animal(not sure if this is correct from a biological standpoint) for example, then the 'including X' part would become important.
Most of these phrases are worded that way to put emphasis on some more common or important cases and generally for more clarity. "not limited to" serves as a reminder to the parties that the list is not exhaustive and it is easier to negotiate with companies outside the court with such terms.
My native language is not English, however when I consider it in my won language we have exact same phrase... And I will have to say that I view this as a linguistic issue rather than legal at this point so here is another link: meaning of the phrase 'Including, but not limited to'