I don't see how you'd pass current through a resistor and prevent it from heating up - if you were oscillating both ends in phase, then you'd never have a potential across it, no current would flow, and you'd have effectively no loss, but if you've got current - AC or DC, positive or negative - you're going to have power dissipation in the resistor.
Just because the average voltage is zero doesn't mean the resistor won't act as a load - the potential would have to be the same on both sides all the time for no power to be lost, but then no current would be flowing through it, and why would you bother putting the resistor in, it would have no effect (in the limited scope of the exercise, of course it could be used for abnormal conditions or fixed impedance at different frequencies or what have you).