In what circumstances would an AC produce no condensate but still cool the air?
Condensate is a byproduct of air conditioning, which is created when moisture in the air condenses on the cold evaporator coils. There are two situations where an air conditioner would not generate condensate.

If there's no moisture in the air, then there's nothing to condense out of the air. 

If the coils are not cold enough (below the dew point), the moisture will not condense out of the air.

Also note, that if the coils are too cold. The condensate will freeze, so it will not drain out until the coil defrosts.

Another note. If the drain does not have a trap,  it's possible that condensate will not drain until the fan shuts off.