Is it true that once a man starts testosterone replacement therapy, he needs to continue it for the rest of his life?

Depends. On the cause. If the cause is a permanent condition such as a chromosomal abnormality or surgical removal of testes...Then u will need replacement for life. If its a functional problem- like recovering from a big illness, or severe insomnia or depression, or prolactinoma, when the problem resolves, the testosterone levels will normalize too, and you wont need to cont testosterone tx.
Typically yes. Usually the indications or reasons to start testosterone require lifelong treatment. There are clear risks and benefits to testosterone therapy. There are risks of stroke, heart attack, and stimulating prostate cancer. The testicles will shrivel (atrophy) and your fertility goes down as well. See your doctor for more information.