Jace, Unraveler of Secrets allows you to create an emblem that will cause the first spell to be countered every turn for your opponents. Does this continue even if he is destroyed? Is there a way to counter or remove its effect?
The Emblem will stay forever, even if Jace is destroyed.
113.1. Some effects put emblems into the command zone. An emblem is a marker used to represent an object that has one or more abilities, but no other characteristics.
113.2. An effect that creates an emblem is written “[Player] gets an emblem with [ability].” This means that [player] puts an emblem with [ability] into the command zone. The emblem is both owned and controlled by that player.
113.3. An emblem has no characteristics other than the abilities defined by the effect that created it. In particular, an emblem has no name, no types, no mana cost, and no color. 113.4. Abilities of emblems function in the command zone.
113.5. An emblem is neither a card nor a permanent. Emblem isn’t a card type.
There is nothing in the game currently that can remove an emblem. (Other than restarting the game with Karn Liberated). There shouldn't be any rule preventing the existence of a card that said something like "Exile target Emblem", but no such thing exists.