Toggles that are suppressed will continue to drain endurance and provide mez protection, but all other attributes will shut off.
In PvE, a suppressed toggle will stop providing all buffs except mez protection. For example, a Force Field Defender is hit with a Rikti Mentalist's Mezmerize. The Defender is now sleeping and her Dispersion Bubble is no longer providing her with any defense. It is still providing mez protection and if the Defender is next hit with a Dominate she will not be held. In this situation, the Defender's teammates will still benefit from the defense and mez protection so long as they themselves remain un-mezzed.
In PvP, defensive toggles do not suppress for any AT.