John Ritter died unexpectedly during the run of his sitcom 8 Simple Rules and the show's producers decided to continue the series. He was the father/husband in a family sitcom (Katey Sagal played his wife), so they had a tough road. It lasted a season or two after Ritter's death, but lost ratings steadily.
Cover Up was an action-adventure series starring Jon Erik Hexum and Jennifer O'Neill. The impossibly handsome and charismatic Hexum died on set in an accident involving a prop gun. The show hired Antony Hamilton to 'replace' Hexum, but viewers didn't stick around.
I'd cite Happy Days continuing when Ron Howard left the series, but Henry Winkler had been taking over the show well before Howard left, so the transition wasn't as hard as it would have been earlier on.
When Cindy Williams left Laverne and Shirley, they went on with co-lead Penny Marshall (Laverne) by herself, but the show limped along stupidly without Williams's Shirley. The missing "half" of the series was most obviously absent in the opening titles, where instead of the ladies singing their funny/nonsensical song as they marched down the street, the producers had some strange children sing it as Laverne, walking alone on the sidewalk, watched them march by.
