- On Married With Children when the dog who played Buck got too old they had Buck die and go to dog heaven, only to be reincarnated and send back to the Bundys as new puppy Lucky.
And actually the Dallas character of Miss Ellie is an example of a “Darren.” Barbara Bel Geddes originally played the character and then left. Donna Reed then played the character until Barbara Bel Geddes returned.
Mrs Kravitz on Bewitched was another instance where the actress died and was seamlessly replaced with another woman (who looked nothing like her).
Sorry, I believe Mental Floss missed the boat on this one by at least 15 years. In 1957, Lassie, then on CBS, starred Tommy Rettig as Jeff Miller, Jan Clayton as Jeff’s mother, Ellen, and veteran character actor George Cleveland as Jeff’s Grandpa, George. George Cleveland died unexpectedly in 1957. The new season opened with Gramps’ funeral and with Jeff and his mother selling the farm to Paul and Ruth Martin and moving away.
Stefano DiMera of Days of Our Lives is a strong contender. His Wikipedia article lists at least 12 “deaths”.
Valerie Harper deserves a mention here: Probably the only title character to be killed off her own show!
Video of part of that here.
For number 2 I offer The Real McCoys where in 1960 Kathleen Nolan as Kate left the show due to a contract dispute and was said to have died.
As for the George Burns example, since George talked to the camera and narrated the show, the switch didn’t even involve him breaking character.
Damn! I remember watching that episode when I was three or four (it says in my Baby Book that I loved Lassie), probably in a rerun (I would have been two in 1957). I was kind of bewildered that the whole cast was changing, except for the dog, and always wondered what happened to Jeff. 
the Millers…what happened to the first kid?
Well if you think about it, Dick Sargent was the replacement for Dick York. Sargent . . . York. Sergeant York.
No, there’s no point to this.
Tommy Rettig and Jan Clayton both wanted to move on from the show, so Timmy (Jon Provost) was brought in early to ease the transition as Ellen Miller’s foster child and he and Jeff became friends. When the Martins bought the farm from the Millers, they agreed to adopt Timmy and let him stay on the farm. When the Millers got ready to leave, Jeff called Lassie to get in the car, but she wanted to stay with Timmy, so Jeff agreed that she could be his dog.
Paul and Ruth Martin were originally played by Jon Sheppod and Cloris Leachman, but they proved not to be popular with viewers and Leachman was supposed to be difficult to work with, so they were recast with Hugh Reilly and June Lockhart.
I know all this because COZI TV runs all of the old Lassie episodes in the afternoon.
along the lines of actors replaced without anything said, the oldest son was played by some guy who was replaced after the first episode. Apparently his film carreer was taking off.
What oldest son from which show? 
Would that be Chuck Cunningham? 
Could be, but doubtful. The first actor to play Chuck Cunningham, Gavan O’Herlihy, was in nine episodes and lasted longer than the second actor, Randolph Roberts, who was written out of the series after only two episodes.
Lockhart was not a Kate replacement but a doctor who came to Petticoat Junction.
The actor who played Jim Garner’s dad in the Rockford Files pilot was replaced without comment by Noah Beery (an excellent choice) when the show went into production.
Weren’t the original Bud and Kelly both replaced after one episode of Married … with Children?
No, they were always played by David Faustino and Christina Applegate.