What will happen if a wire that is plugged in heats up?

its casing will dry out, crack and catch fire, or it will crack off and cause arking, which can cause a fire... ultimately, check the gauge size of the circuit and make sure it corresponds to the proper amperage of breaker, if this is all good, you need a bigger cord(the gauge size must be sufficient for the intended tool/device.. a drill that has a 16 amp rating needs a 12 gauge cord whereas a 12 amp drill needs only a 14 gauge cord.