The characters are the heart and soul of a television show. The viewers develop real connections with the characters in their favorite shows, they care about them, and might even write some creepy fan fiction. That is why, when a character is replaced, fans will riot, and the show will usually take a turn for the worse.
The first show that comes to mind would be The Office. Steve Carrell left after a very successful six seasons and was replaced as regional manager by Ed Helms as Andy Bernard. Out of all the characters on The Office, Helm's personality and mannerisms are the most like Carrells. They share the same lovable offensive qualities and both strive for attention and acceptance from their superiors and coworkers. The sixth seasons finale featured an array of guest stars all vying for the position of manager. However, losing a newcomer and putting Andy in charge was a very safe move by the writers in order to keep up the same pace of the show.
Another example is the show Misfits, which premiered it's third season this Sunday minus one of the more memorable characters. Robert Sheehan played the witty, Irish comic relief on the show but left after the second season to pursue more movie roles. Channel 4 put out a short on their website to explain his absence and quickly introduce his replacement on the show. Again, what we experience of Rudy, the newcomer, is a glimpse of a similar character. He seems to slide into the role of comedian and troublemaker, as was his predecessor Sheehan.
The same can be said about Doctor Who, especially since that show goes through actors every two to three seasons.. Every time the doctor regenerates and is recast, there is always a divide between the fans. Mainly in the new series, there is a solid amount of people who see David Tennant as the real Doctor, even though he hasn't been on the show for two seasons, being replaced by Matt Smith. People will always have their favorites, and stand behind them as the true actor for that role.
Luckily these example happen to have an extremely good supporting cast that can easily support the show after their stars depart. A very common theme throughout this seems to be that the essence of the character remains the same, to make it easier for different actors to filter through the role.