I don't know all of them but here are a few schools and the animals they have:
Air Force Academy - falcon named "Yeti"
Arkansas - razorback (wild boar) named "Tusk"
Baylor - They have a live bear habitat on campus:
Colorado - Ralphie the Buffalo
Colorado State - Cam the Ram
Connecticut - A husky named "Jonathan"
Georgia - Probably the world's most famous bulldog "UGA"
Georgia Southern - "Freedom" the live Bald Eagle who flies to the field before kickoff and is escorted along the sidelines and in the stands by his handler.
Yale has a live bulldog named "Handsome Dan"
Georgetown also has a bulldog named "Jack the Bulldog"
Houston had a live cougar until 1989. I guess they had difficulty controlling the last one so they stopped using live cougars.
LSU - a live tiger named "Mike"
Memphis - a tiger named "Tom"
Navy - a goat named "Bill the Goat"
North Alabama - 2 lions named Leo & Una
Oklahoma - OU has 2 white ponies named "Boomer" and "Sooner" that pull the "Sooner Schooner" (a covered wagon)
South Carolina - rooster named "Sir Big Spur"
SMU - a black stallion named "Peruna"
Interesting note, Peruna led the SMU band in President Bush's inauguration parade.
Texas - a Longhorn named "Bevo"
Texas A&M - a collie named "Reveille"
Washington - an Alaskan Malamute named "Spirit"