The Short Version
There is a real Chinese law (2007 Order No. 5) requiring government approval for the recognition of reincarnated "Living Buddhas" (tulkus) in Tibetan Buddhism — but the claim overstates it in two important ways. First, it applies only to Tibetan Buddhist tulku lineages, not all Buddhist monks in China. Second, the law governs the official recognition and management of reincarnation successions, not literal permission for a person to be reborn. The underlying regulation is genuine and enforceable, but the claim's broad wording gives a materially misleading impression.