Remains of Tinirau were first discovered by paleontologist Joseph T. Gregory in 1970, who then worked for the University of California Berkeley. Gregory and his field team found these fossils near the Simpson Park Mountains in Eureka County, Nevada. They came from a deposit called the Red Hill I beds, which date to the upper Givetian stage of the Middle Devonian. The Red Hill I beds include a series of limestone and mudstone, likely deposited in an outer continental shelf marine environment. Six fossils are known, all preserving bones of the skull. Two fossils preserve postcranial bones in articulation. The holotype specimen, UCMP 118605, is a mostly complete skeleton.