Driving LCD segments with an LC generated sine wave to save power?
A series capacitor would get rid of the DC bias issue. 

In theory (ignoring inductor resistance and hysteresis losses in the LC fluid etc.) resonating the capacitance with an inductor would work (only taking energy when the state of the segment changes) but I doubt it makes sense practically. Even if a segment was as high as 100pF it would take thousands of H to resonate that at a reasonable frequency. 

Simply switching the segments means a loss proportional to \$fCV^2\$ so reducing f, decreasing C (smaller segments) and reducing the voltage will minimize power consumption.