|Another Time, Another Place (1958) [Mark Trevor]: Killed in a plane crash (off-screen); we learn of his death when Lana Turner hears the news on the radio.
The Crucible (1959) [John Proctor]: Presumably executed by hanging. (I haven't seen this version.)
Tarzan's Greatest Adventure (1959) [O'Bannion]: Shot with an arrow by Gordon Scott.
Macbeth (1961) [Macbeth]: Stabbed to death in a swordfight with Woodrow Parfrey.
From Russia with Love (1963) [James Bond]: Playing a dual role as both Bond and a double, the double is garroted by Robert Shaw during a training exercise, in preparation for his planned assassination of the real Bond. (Thanks to Garrett)
Woman of Straw (1964) [Anthony Richmond]: Falls to his death when Johnny Sekka throws a wheelchair at him.
The Red Tent (Krasnaya palatka) (1970) [Roald Amundsen]: Frozen to death after being stranded in the ice while attempting to rescue Peter Finch.
Zardoz (1974) [Zed]: Dies of old age, along with Charlotte Rampling. Shown in a time-lapse sequence with heavy make-up.
The Man Who Would Be King (1975) [Daniel Dravet]: Falls to his death in a ravine when the natives cut the ropes holding up the bridge; his mummified severed head is shown afterwards when Michael Caine presents it to Christopher Plummer. (Thanks to Justin)
The Arab Conspiracy (Double Hit; The Next Man) (1976) [Khalil Abdul-Muhsen]: Killed by Cornelia Sharpe.
Robin and Marian (1976) [Robin Hood]: Poisoned by Audrey Hepburn, before she drinks the poison herself; the movie ends with the dying Sean firing one last arrow into the air. (Thanks to Gary)
Sword of the Valiant (Sword of the Valiant: The Legend of Sir Gawain and the Green Knight) (1984) [The Green Knight]: Killed (though I don't know the details). (Thanks to John)
Highlander (1986) [Juan Sanchez Villa-Lobos Ramirez]: Decapitated with a sword by Clancy Brown. (He was brought back in the 1991 sequel Highlander II: The Quickening.) (Thanks to Uwe)
The Untouchables (1987) [Jim Malone]: Machine-gunned by Billy Drago in an ambush after Sean chases another thug out of his house; he dies shortly after Kevin Costner arrives at the house. (Thanks to Gary, Rommy, Neil, and Tal)
Family Business (1989) [Jessie McMullen]: Dies of natural causes. (Thanks to Gary)
A Good Man in Africa (1994) [Dr. Alex Murray]: I vaguely recall that he died in this movie, though I can't remember the details.
First Knight (1995) [King Arthur]: Shot in the chest with an arrow by Ben Cross. (Thanks to Matt)
Dragonheart (1996) [Draco]: Providing the voice of a dragon, he is stabbed in the heart by Dennis Quaid in order to kill David Thewlis thorugh their mystical connection. (Thanks to Dave and Derek)
The Avengers (1998) [Sir August de Wynter]: Struck by lightning from his weather-controlling device (levitating him into the air in the process), after being stabbed in a swordfight with Ralph Fiennes. (Thanks to Neil)
Finding Forrester (2000) [William Forrester]: Dies (off-screen) of cancer; we learn of his death afterwards when Matt Damon informs Rob Brown. (Thanks to Neil)
The League of Extraordinary Gentlemen (2003) [Allan Quatermain]: Stabbed in the back by Richard Roxburgh; he dies shortly afterwards. After he is buried in Kenya, a witch doctor is shown performing a ritual, suggesting that Sean is about to be brought back to life. (Thanks to Gary and ND)
Indiana Jones and the Kingdom of the Crystal Skull (2008) [Dr. Henry Jones Sr.]: Although Sean does not actually appear in this movie, it's revealed that his character from Indiana Jones and the Last Crusade had died off-screen between the two movies. (Thanks to Tommy, Andrew, Liz, and Augustin)