No. If the voltage is the same as the transformer normally gets, it should run cooler than if it had a load on the secondary.
A transformer without a load on the secondary is just an inductance. So, the current in this is determined by the supply voltage, the frequency and the inductance and resistance of the primary winding.
Many appliances switch power in the secondary circuit so that the primary is always connected to the mains. This does waste some power (compared with turning off the primary) and it also leaves the primary vulnerable to mains voltage surges.