|Monty Python's Flying Circus: Episode 1, Whither Canada? (1969): (1) [Genghis Khan] Dies of a heart attack; we see him pacing around his tent when he suddenly shrieks and flings himself backwards without warning. (2) [Doctor] Laughs himself to death after reading the world's funniest joke while examining Michael Palin; we hear him laughing from outside the house, then see him fall halfway out of the window in the background. (3) [Gestapo Officer] Laughs himself to death after Michael Palin recites the world's funniest joke in German in the interrogation room. (Note: The Pythons played multiple roles in each episode; I will only be listing the specific characters that die.)
Monty Python's Flying Circus: Episode 5, Man's Crisis of Identity in the Latter Half of the 20th Century (1969) [Man in the Street]: Hit by a car while standing in the road for a "man-in-the-street" interview.
Monty Python's Flying Circus: Episode 11, The Royal Philharmonic Orchestra Goes to the Bathroom (1969): (1) [Inspector Tiger] Shot in the head, shot in the neck with an arrow, and poisoned (all off-camera) when the lights go out; his body is shown when the lights come back on. (2) [Undertaker] Dies of exhaustion while carrying a heavy coffin with three other undertakers; the others then put his body in the coffin and continue carrying it.
Monty Python's Flying Circus: Episode 12, The Naked Ant (1970) [Nigel Incubator-Jones]: Commits suicide by shooting himself in the head at the end of the Upper-Class Twit of the Year Competition.
Monty Python's Flying Circus: Episode 13, Intermission (1970) [TV Interviewer]: Shot in the back by Eric Idle while presenting a TV show.
Monty Python's Flying Circus: Episode 17, The Buzz Aldrin Show (1970) [The Reverend Neuk]: Killed in an explosion when he prepares to baptize a "baby" which is actually a disguised bomb, just as Terry Jones arrives too late to warn him.
Monty Python's Flying Circus: Episode 21, Archeology Today (1970) [Sir Robert Eversley]: Killed (along with Carol Cleveland) in a fight with Terry Jones and Michael Palin at an archeological dig site, while Carol is standing on John's shoulders; their bodies are shown lying on the ground afterwards while the dying Michael speaks to the camera.
And Now for Something Completely Different (1971): (1) [Announcer]: Possibly killed in an explosion after announcing the title; he reappears several times afterwards (including one scene where he is being roasted on a spit for another possible "death"), so this may not count, but I thought I'd list it just in case. (2) [Vivian Smith-Smythe-Smith]: Accidentally shot in the stomach by Eric Idle during the Upper-Class Twit of the Year Competition.
Monty Python's Flying Circus: Episode 30, Blood, Devastation, Death, War and Horror (1972) [Terence Rattigan]: Shot in the neck with an arrow by a pantomime goose.
Monty Python's Flying Circus: Episode 33, Salad Days (1972) [Pianist]: Bleeds to death when his hands are severed by the piano lid closes on them.
Monty Python's Life of Brian (1979): (1) [Stoning Official]: Crushed to death when a group of women drop a giant rock on him during a stoning, after he says "Jehovah" while explaining that the stoning hasn't started yet. (2) [Crucifee]: Executed by crucifixion; the movie ends with him and the other prisoners singing as they hang on their crosses. (Thanks to Katherine and Tommy)
Monty Python's The Meaning of Life (1983) [Sturridge]: Shot to death by German soldiers while John is getting a tablecloth to serve Terry Jones' cake in the trench. (All of John's other characters survive the movie.) (Thanks to Tommy)
Yellowbeard (1983) [Harry 'Blind' Pew]: Killed in an explosion after Peter Boyle and Marty Feldman lock him a shed with a bomb. (Thanks to Anton)
Mary Shelley's Frankenstein (1994) [Professor Waldeman]: Stabbed in the stomach by Robert De Niro when John tries to give him a smallpox vaccination.
Harry Potter and the Sorcerer's Stone (Harry Potter and the Philosopher's Stone) (2001) [Nearly Headless Nick]: Plays a ghost who was executed by a botched beheading centuries ago; he reprises this role in some of the sequels, but I'll conserve space by only listing this first film.
Monty Python's Personal Best: John Cleese's Personal Best (2006) [John Cleese]: Dies of a heart attack while going off on a racist rant during an interview with Dayna Devon. (Played for comic effect.) (Thanks to Tommy)
Shrek the Third (2007; animated) [King Harold]: Providing the voice of a frog, he dies of old age/illness (after numerous "false alarms" of going in and out of consciousness), with his family gathered by his bedside. (Played for comic effect.) (Thanks to Eric)
Igor (2008; animated) [Dr. Glickenstein]: Killed in an explosion when his rocket malfunctions. (Played for comic effect.) (Thanks to Tommy)