Ponderomotive Effect on Ionic Liquids Would an alternating electric field cause net motion of ions in a liquid in the case of the electrodes being placed outside the liquid?

Generally speaking, what happens if you apply an external electric field to an ionic liquid, taking into account that there are no electrodes places inside the liquid itself, but are separate from the liquid container in the same manner you have a parallel plate capacitor.
Yes, due to electrostatic coupling. The conductors and the liquid would form a capacitor, causing back-and forth motion of the ions (and in polar molecules). This is an effective way to heat something without contact, as in dielectric heating.

This capacitance enables liquid crystal displays and E-Ink to display images using little or no current when not actively switching.