I just got a brand new cell phone. Overnight, it obviously fully charged. Will I harm the battery if I let it on the charger even if it is fully charged?

Read the instructions that came with the phone. Some instruments don't like living on the charger, some don't care.

Sometimes the charger has a shut off when it is fully charged, just for safety. Did it get hot, if so you might have damaged it. Like es said read the instructions, sometimes they can actually be helpfull :D