If a 16-year-old man has a hard lump inside his nipple does that mean he has cancer?

Not necessarily. Hormonal changes in this age group can cause gynecomastia which is swelling of breast tissue. See your doctor for an examination to confirm.
No. It is normal to feel a small lump of breast tissue behind the nipple in a teenaged boy. This can become enlarged during or just after puberty and is called gynecomastia. Breast cancer is unheard of in teenaged boys.
Transient change. About a half of males will undergo a transient nipple/breast enlargement at the onset of puberty. Normally it will resolve slowly as other hormonal changes kick in over about a year. Less often it can persist over 3 years and surgery may be needed for cosmetic improvement. Carrying around extra weight can aggravate the issue as fat can masquerade as breast tissue.