Is it true that surgery is required in order to remove anal warts or can anal warts disappear on their own?

Not always: Almost all warts -- common warts of the hands and feet, genital warts, anal warts -- are eventually cleared by the immune system. They'll probably resolve, but it seems to take longer, often several years, for anal warts. If many warts or especially large ones, infection and other complications are a risk and surgery usually is wise. Just a few warts can be treated by freezing, imiquimod, etc.