It depends on the artifact.
Continuous ones (like Kings Sword or Altar of Incense) have their abilities active so long as they are active (face up)
Triggered ones (such as Captured Ark or Ark of the Covenant) only get used when their trigger is met. So for Ark of the Covenant if the condition is met (following a rescue attempt) the discard happens.
Finally Optional Trigger abilities (like Unholy Writ) have the ability saying something like "You may do something to do something else" or alternatively "at any time..." (although the only card that I can think of that says that is Gates of Hell, which is a fortress). These ones you can use so long as any conditions on the card are met (and it is active)
Basically a good rule of thumb is that if the card says you may do something then it is an optional ability if it does not, so long as the conditions are met it will active.
