Can you puncture a lung by lifting something very heavy?

Probably not. A "punctured lung" with no obvious cause is a spontaneous pneumothorax, which is a tiny air leak through the outer lining around the lung. It may happen if there is a sudden increase in the outside air pressure, such as being close to an explosion. The air pressure might go down the throat, into the lungs, and push air through a weak spot in the lining. Lifting doesn't cause that kind of pressure.
No. But if you have emphysema, a bleb in the lungs may rupture creating a pneumothorax.