Unless your amplifier has instructions that tell you not to turn it on without a speaker and/or input load present, I would start by taking the amplifier and plugging it into the Variac, with nothing connected to the inputs and nothing connected to the outputs of the amplifier. Make sure the amplifier and Variac are turned off when you do this and that the Variac is set to minimum voltage.
I would then hit the power on button/switch for the amp, wait a couple of seconds and then turn on the Variac.
I would raise the Variac voltage every couple of minutes by 25 volts, if there is no indication of a problem (no noise, smoke or burning smell). When you have it all the way to full voltage I would let it sit there for an hour. If all seems fine, power off the amplifier, turn off the Variac and turn the voltage on it down to zero.
If at any point in time something appears to be going wrong I would power off the Variac immediately.
If all appears well with the amplifier after an hour, I would unplug the amplifier form the Variac, I would then connect a source and speakers to the amplifier and plug the amplifier back into the Variac.
I would then plug the amplifier into the Variac, press the power button on the amplifier, switch on the Variac, and slowly turn up the voltage. Again if anything appears to be going wrong power off the Variac immediately, unplug everything and take the amplifier to a repair shop for evaluation.