Can the ions of dissolved salts be physically separated with a strong enough external electric field?
Could an external electric field (e.g. the electrically charged plates in Millikan's oil drop experiment) physically pull apart the dissolved ions in the water if it was strong enough?
What forces would it need to overcome?

Yes, this is the principle behind electrophoresis and electric double layer supercapacitors where charge is separated by an electrostatic field shown below.

Diagram of Double Layer Capacitor charging and Discharging

What would be the outcome if the $\ce{Na+}$ and $\ce{Cl-}$ ions were then physically separated into two different containers?

Assuming you could so this without any interaction with the container, you would have 2 Leyden Jars that were opposite in charged and storing a lot of energy.