More TV: Freddie Prinz, who commited suicide while on Chico and The Man.
Brenda Benet commited suicide while on Days of Our Lives.
And Phil Hartman of News Radio and The Simpsons 
Tyrone Power died while filming a dueling scene in Solomon and Sheba. I couldn’t find an exact account, but I recall reading that he died in his trailer, not on the set.
Bruce Lee died before he could finish filming Game of Death.
Roy Kinnear died in hospital after falling from a horse on location for The Return of the Musketeers.
No, but he’s not at all well. 
World famous stunt pilot Art Scholl crashed and died while doing footage for Top Gun.
Similarly, pilot Paul Mantz was killed while flying the plane for the 1965 Flight of the Phoenix.
Didn’t the original Wicked Witch of the West die in The Wizard of Oz?
Although Phil Hartman’s death was indeed tragic, it didn’t happen on-set: he was murdered by his wife at home.
If TV shows count, John Ritter collapsed on stage during a taping of an episode of Eight Simple Rules for Dating My Teenage Daughter. He was pronounced dead at the hospital.
I know it’s morbid, but does anyone else wonder how long it took them to turn the cameras off?
There have been cases in which comedians have died onstage (literally) and the audience, believing it to be part of the act, laughed and anticipated what would happen next. Since Ritter was known for his pratfalls on Three’s Company, I wonder if the audience thought it was a joke or part of the script.
I’m also reminded of an episode of The Dick Cavett Show which never aired due to an unusual circumstance: during the interview, one of the guests appeared to fall asleep sitting down and started to “snore.” Cavett jokingly asked the guest, “are we boring you?” There was no response: he had died. I’m not sure if they kept taping after this happened.
Trinidad Silva was killed by a drunk driver (not on set) before finishing all his scenes in UHF.
Supposedly, when Redd Foxx died on set, everyone thought he was faking. Ditto with British TV comedian Tommy Cooper.
No.
Both Margaret Hamilton (who had the role) and Gale Sondergaard (whom she replaced) died in 1985, well after the movie was finished.
Zev Steinhardt
Gale Sondergaard was offered the part of the Wicked Witch of the West, but she backed out because she didn’t like the grotesqueness of the makeup. She had hoped to be a more glamourous witch. She didn’t die until 1985.
Dick Shawn (from the Producers and It’s A Mad, Mad, Mad, Mad World) died on-stage of a heart attack. The audience, IIRC, thought it was part of the act.
Zev Steinhardt
Another live TV death was Tommy Cooper the British comedy magician. As the link says “On 15th April 1984, Tommy Cooper collapsed during a live television broadcast at Her Majesty’s Theatre. The people in the audience were still laughing as Tommy, lying on the stage, had the curtain brought down on him and the show cut to a commercial break.”
I thought he died during a rehearsal, in which case the cameras probably were not rolling.
Steve Irwin, of course, died while filming one of his nature specials.
Jean Harlow died during the filming of Saratoga (1937) and an unconvincing body double was used in several scenes that hadn’t been shot before Harlow’s death.
Oliver Reed died during the filming of Gladiator (2000.) IIRC, his digitized face was electronically pasted onto shots of a stand-in’s body, and another actor dubbed the voice.
Aaliyah died after the principal filming of Queen of The Damned. There aren’t any visual clues since filming was done but you can hear it because they called in her brother for voice dubbing during post production.