Though I haven't personally visited yet, the Steinhardt Museum of Natural History which is on the campus of Tel Aviv University in Tel Aviv, Israel has live invertebrate displays inside it's museum building. Those include several stick insects, hissing cockroaches, lubber grasshoppers, a nest of Camponotus, Gryllotalpa, crickets, grasshoppers, assassin bugs, scorpions, and a few spider species etc. The collections manager in their entomology department is friend of mine. I actually thought about sending him some specimens of Arizona native insects I have stored in my freezer. He doesn't oversee the live invertebrate displays, another curator and collection manager deal with that.
Meir Segals Garden University Zoo which is situated next to the museum building is actually a part of the museum, similar to Ralph Perkins II Wildlife Center & Woods Garden at the Cleveland Museum of Natural History. Like the wildlife center at the Cleveland Museum of Natural History, Meir Segals Garden University Zoo specializes in taxa native to its region, but that is not all they keep. Native Israeli/Palestinian species and subspecies kept at Meir Segals include Syrian rock hyrax, Palestine blind mole rat, Syrian striped hyena, Palestine jungle cat, Syrian golden jackal, Anatolian wild boar, Mesopotamian fallow deer, Palestine mountain gazelle, Nubian ibex, Palestine red fox, Tristam's jird, Sundevall's jird, Long eared hedgehog etc.
As far as natural history museums with live animal exhibits that I have personally visited, boy do I have a lot to contribute! I will share such examples in further posts I plan to make in this thread.