One possibility is that it is a shortened form of the Germanic personal name "Biher", which is composed of the elements "bi" (meaning "bee") and "heri" (meaning "army" or "warrior").
Another possibility is that it is derived from the Old High German word "bihhira", which is a term for a beekeeper.
The Bihr last name is found in many European countries, including Germany, Switzerland, France, and the United States.