Legend has it that he has become an immortal, but there is no way to prove it (theoretically, he can become an immortal, personal opinions, such as cracking the subconscious mind's control over the body and mastering the laws of the universe). Studying him all his life, even with the experience of his predecessors, who can do it? Laozi's Tao Te Ching is a Taoist Bible, which can be learned and understood. They must get something or feel something by doing so. Otherwise, it wouldn't exist for so long.
Whether Lv Dongbin is immortal or not depends on your understanding and view of Taoism.