Why a transformer designed for a 120V AC input will often burn out if connected to a 120V DC source?

All Transformers are designed to work on AC. They do not work on DC.

If you connect an inductor to DC, the current will increase until the capacity of the source or the conductance (1/resistance) capacity of the inductor and conductors is reached. Often, this condition will overheat and destroy the inductor, or destroy the source. A transformer is not an exception, as it is a form of inductor.