Can manually throwing the dryer circuit breaker cause it to burn out?
Manually flipping a breaker, especially without a significant active load, should never melt a breaker.

A melted breaker indicates a large heat build-up. This could be a defective breaker but is more likely a bad connection, producing resistance, which heats up under normal load. This could be an insufficiently tightened screw connection to the hot wire, poor connection from the breaker to the panel, or such. It could also be a breaker not rated for the panel, producing a poor connection.

First question: do you have aluminum wiring in your house? If so, the circuit breaker must be rated for use with aluminum and the wire should be properly treated before connecting. Improper treatment of aluminum wiring has caused many house fires.

Second question is what brand is your breaker panel? There are several brands from the 60's and 70's that are generally acknowledged as fires waiting to happen.

In any case, a melted breaker indicates a serious hazard! I highly recommend getting a professional electrician in to look things over!