Well, for one running things on a lower voltage doesn't necessarily mean a higher current draw. For instance, resistive loads like incandescent lights and electric heaters would draw less current at a lower voltage. Things like switching power supplies would attempt to compensate for the lower voltage by drawing more current.
However, this isn't entirely relevant as you mentioned a key component: the inverter. The inverter will convert your 24v or 48v DC into 110v AC to power any connected AC devices. The waveform coming out of the inverter is a decent approximation of what comes from the electrical grid, so the devices should behave normally and draw the same amount of current that you would expect if they were connected to normal AC line power. Where you will see an increase in current draw is at the input to the inverter, but the system would be designed to handle this.