|The Tragedy of Richard III (1983) [Lady Anne]: Killed (off-screen) by Ron Cook. (Thanks to Gordon)
Othello (1990) [Emilia]: Stabbed by Ian McKellen; she dies shortly afterwards while talking to Willard White.
Shakespeare: The Animated Tales: Macbeth (1992; animated) [Lady Macbeth]: Commits suicide (presumably off-screen).
Gormenghast (2000) [Clarice Groan]: Dies of starvation (off-screen), along with her sister (Lynsey Baxter), after Jonathan Rhys-Meyers locks them in a room and abandons them; their skeletal bodies are shown afterwards when Jonathan returns to the room much later.
Doctor Who: New Earth (2006) [Cassandra]: Providing the voice of a disembodied face (on a sheet of skin connected to a brain in a jar), the brain dies after her mind is transferred into Billie Piper's body. After her mind is transferred again into Sean Gallagher's body, she finally dies when Sean's body reaches the end of its artificial lifespan, after she/he goes back in time and meets herself in her original body; she thus dies in her own arms. (I had previously listed her apparent death in the 2005 episode The End of the World, but this episode revealed that it was only her skin-sheet that was destroyed then; her brain survived until this episode.)