Why do we need to remove flux from circuit boards?
If your boards contain high-impedance analog circuits, then the conductivity of the flux is a real concern. Leakage current through flux is a common source of error in high-gain high-impedance analog circuits.

For other types of circuits, reliability is a bigger concern. Fluxes are reactive chemicals, and if left on the board they can cause corrosion and lead to circuit failures in the field. There are "no clean" fluxes that are meant to minimize this issue, but even these might not be appropriate for high-value circuits with high reliability requirements.