Does a 16 month old infant need a passport to travel to Jamaica?

Not if they live in Jamaica. Otherwise yes.

But that's not always exactly true. I just went on a cruise that stopped in Ocho RiOS, Jamaica. They told us we'd need our passport to get off and on the ship. But no one ever stopped us to ask for one. I had three people offer to sell me pot in about 5 minutes, but no passport control...