This answer will, by necessity, contain spoilers for the mentioned story arcs:
In the comics, there were several attempts on the life of The Joker, usually unsuccessful ones planned by other DC villains. However, there were several cases (in various media) where The Joker was actually killed:
Comics
In the ground-breaking Kingdom Come series, The Joker traveled to Metropolis and used his poison gas to kill everyone in the Daily Planet (including Lois Lane). While he was later surrendering to the police, a vigilante named Magog rushed up to the scene and summarily executed The Joker on the spot in front of a shocked Superman. Magog's acquittal for the act was the event that caused Superman to leave Metropolis and enter a self-imposed exile at the Fortress of Solitude.
In the classic Detective Comics #42 story "The Joker Walks the Last Mile" (1942), the villain surrenders of his own accord and is actually executed for his crimes by the court system. He is later revived, however, so that one may not count. This story was later re-used in Batman #5.
Video Games
In the parallel Earth of Injustice: Gods Among Us, an enraged Superman killed The Joker with his bare hands after the villain taunted him about killing his wife and son. After crossing a line he swore he'd never again cross, Superman then gave up on his principles altogether and swore to make the world safe by killing ALL villains.
In Batman: Arkham City, Joker is revealed to be dying from his previous exposure to the Titan formula, and exposes Batman to the same disease. After obtaining and using the cure, Batman debated whether or not to cure Joker with it. The vial was destroyed when Joker attacked Batman, so technically it was his own actions that caused his death. However, the fact that Batman had the cure at hand and didn't use it could count as being implicit in the death. (Thanks to @MissouriSpartan for the update.)
Cartoons/Film
In the un-edited version of Batman Beyond: Return of the Joker, it is revealed via flashback that The Joker was shot dead by Tim Drake. This was a direct result of the torture, brainwashing, and general horror that the villain had put him through during his capture days before the event. In the edited release of the film, young Drake instead pushes The Joker into an electrical panel, thus electrocuting him (thanks to @Axelrod for the update).
Executions by Batman
Also, in addition to the one you mentioned in your question, there are at least two other instances in which Batman himself decided to finally do something about The Joker:
In the Elseworlds arc JLA: The Nail, Batman himself finally decided to end the villain's career and savagely beat him to death before breaking his neck. This may have been a nod to the events of The Dark Knight Returns, in which Batman made the same decision, beat The Joker savagely, yet stopped himself after partially breaking the villain's neck. A spiteful Joker then finished twisting his own neck in order to frame Batman for the killing.
In the 1989 live-action film, The Joker plummeted to his death after Batman anchored his leg to a stone gargoyle using steel cable. It's debatable, however, whether or not Batman actually intended to kill him or simply sought to prevent him from escaping.
Conclusion
Obviously this list isn't exhaustive, as there are likely numerous other examples where the Joker was killed by the actions or inactions of someone in the name of justice. However, as you can see, such examples definitely DO exist.
