It's a little bit complicated. For a purely resistive load, yes power consumption will in theory increase with an increase in voltage. Energy is power x time, so energy consumption also increases. However the resistance of a purely resistive load will increase with temperature, so energy will be less than it would have been if the resistance stayed the same.
For other loads, power consumption may stay the same. E.g., switch mode power supplies used for charging power tool batteries or powering notebook computers are often dual voltage and work on 110 or 230-volt supplies. These don't use more power on the higher voltage because electronics in the chargers regulate the op and take less current from the mains as voltage increases.