DxM cards are always discarded after use unless the card says otherwise or their card type is said to stay in play in the rules (such as Races, Classes and Monster Enhancers).
There are two types of continuing Curses.
- Those that effect an event in the future, such as your next battle.
- Those that say they stick around until something happens (or a Wishing Ring is used), or they have an effect that lasts until the Curse is removed, such as Curse! One Hand Tied Behind Your Back!
Most Curses will simply have you discard cards, or Change Race, and seeing as the effects are completely resolved, they aren't continuing and you can discard them.