The entire FOX television network has been called the spiritual successor to the Du Mont network, and not just because it's the "fourth network" in the American TV lineup — after DuMont went bust in 1956, the two remaining owned-and-operated stations (WABD and WTTG) formed the DuMont Broadcasting Corporation, which grew and became the Metropolitan Broadcasting Company in 1957 and Metromedia in 1961; in 1986, Rupert Murdoch bought Metromedia's television operations and used them to launch the FOX network.
Oh, and the Fox Broadcasting Center is right where DuMont flagship WABD (now WNYW) sits — the former DuMont Tele-Centre, later called the Metromedia Telecenter during that era.
The vast majority of fans consider Lincoln Heights to be the successor to The Cosby Show, and for good reason. Lincoln Heights is essentially the Cosby Show had Cliff and Clair not gone to medical and law school.
Pair of Kings could be considered as one to Wizards of Waverly Place as they're both fantasy comedy with action elements and some grim world elements here and there.
In turn, Wizards of Waverly Place is to Phil of the Future, in the sense that it's Disney's current show with supernatural/magical elements. It's been said that the money Disney gained from the success of Phil was used to create Wizards.
The Prisoner may be considered a Spiritual Successor to Danger Man. Patrick McGoohan plays the same type of secret agent character in both. Some fans (and George Markstein, one of the co-creators of the series) go farther, arguing that Number Six is John Drake, which would make it a true sequel series rather than a Spiritual Successor. However, McGoohan (the other co-creator) denies this, and character differences between Number Six and John Drake call it into question as well. For more details, see the "John Drake?" section of The Other Wiki's article on Number Six.
The Bill was a spiritual successor to The Sweeney. It was made by the same production house (Thames Television), and in its very earliest years it even shared some of the same production team (in particular original executive producer Lloyd Shirley).
The Sweeney was actually made by Thames subsidiary Euston Films, and is closer in style to Euston's earlier Special Branch.
It could be said that "Drake & Josh" was much more a spiritual successor of The Amanda Show, to the point of being a very indirect spinoff. Drake and Josh were regular cast members on The Amanda Show. The Amanda Show itself, and the aforementioned Kenan and Kel, were both spiritual successors to All That.
Jam is this to Big Train - made by the same people, featuring the same actors and using the same general sketch style. The only difference is that the humour in Jam is far darker (due to Chris Morris being its main creator).
While BTR might be a successor to 2gether, the short-lived MTV sitcom which featured a fictional boy band, that pretty much parodies the popular boy bands of the late 90's, which it seems to also do. You can admit that Big Time Rush is almost a Take That and Affectionate Parody to those types of boy bands, whereas 2Gether was just a flat-out Take That. Especially considering they basically revived the genre. Not saying that's a good thing, but YMMV.
Doctor Who. The UNIT stories clearly draw from Quatermass, with the Third Doctor taking on the Quatermass role of a cantankerous scientific advisor aiding military authorities to repel aliens invading Britain, while hampered by the occasional Obstructive Bureaucrat. In fact we're first introduced to The Brigadier in "The Web of Fear", which, like Quatermass and the Pit, featured an alien menace in the London Underground.
The 2009 Easter special, "Planet of the Dead", had a direct Shout-Out by having a geeky UNIT scientist name a unit of measurement after Bernard Quatermass.
The West Wing was born of material and ideas left over from creator Aaron Sorkin's movie The American President.
Studio 60 on the Sunset Strip was a spiritual successor to The West Wing. In fact, the show's biggest problem was often cited as the fact that the style and tone that worked so well for a show about running the government of the United States felt hilariously out of place on a show about running a Saturday Night LiveExpy.
Studio 60 actually works better as a spiritual successor to Sports Night, being about the running of a TV show. Many jokes and references from the former are either very similar to, or directly taken from, episodes of Sports Night. Ditto for The Newsroom.
In a similar vein, MADtv was a spiritual successor to In Living Color, being Fox's competitor to Saturday Night Live with there being only a year between the two. In Living Color is known for giving rise to the careers of Jim Carrey, Jamie Foxx, Jennifer Lopez, and Shawn and Marlon Wayans, while MADtv arguably did the same for Orlando Jones, Phil Lamarr, Nicole Sullivan, Alex Borstein, Bobby Lee, and current SNL featured player Taran Killam.
Choujinki Metalder was produced by Toei to be a 1980s version of Jinzou Ningen Kikaider. Both involved robotic superheroes with a two-tone color scheme on their body (blue on the left side and red on the right), both end their names with "der" ("Kikaider" literally means "Machineder"), their human forms are modeled after the likeness of their creator's dead son (Jiro's likeness came from Taro's, while Ryusei Tsurugi's was from Tatsuo Koga), and they're both assisted by a rival cyborg in black who formerly worked for the enemy (Hakaider and Topgunder).
The Wire is generally seen as a spiritual successor to the earlier Baltimore police series, Homicide: Life on the Street, which was based on Wire creator David Simon's book and for which he wrote.
It's also a successor to another of Simon's shows, The Corner.
Pushing Daisies is the third (unsuccessful) series created by Bryan Fuller and is seen as a sucessor to both Dead Like Me and Wonderfalls.
Several characters (portrayed by the same actor/actress) have appeared from one show to the other, completing the belief that all three shows exist in the same 'universe'.
Not exactly. The universes where clearly different, with Pushing Daisies being retro-cute, Dead Like Me being fairly normal (save the supernatural elements) and Wonderfalls somewhere in between. All of them do carry themes of fantasy, uniqueness, life and death, golden retrievers and touching people.
Most of Jack Webb's later series (e.g., Adam-12, Emergency!, Project UFO) can be considered spiritual successors to his own Dragnet. They all share a basic approach — following the professional lives of dedicated public servants, filmed in the style of a Police Procedural. In the case of the first two, they also share a universe with Dragnet.
Sliders is a Spiritual Successor to Quantum Leap. The shows share a similar episode forumula, Sliders was advertised at least once as "Quantum Leap with an edge," and dialogue in a later episode implies that Maggie Beckett may be Sam Beckett's niece.
Cheers is a spiritual successor to Fawlty Towers and was actually written as an American version of the selfsame. It was only when the writers realized that the overwhelming majority of the plots were taking place in the hotel bar that they cut the hotel out and just set the show in the bar.
How I Met Your Mother is very much a spiritual successor to Friends, which is itself a spiritual successor to Cheers. All three are sitcoms with some form of restaurant as their default set (Friends is the odd man out, using a cafe instead of a bar), and are centered around a group of unmarried ~30-year-olds, with sexual tension amongst the group.
It's also something of a spiritual successor to The Wonder Years, using the framing device of an older narrator telling the audience stories about his past. How I Met Your Mother took the idea and instead of the fairly straightforward application in The Wonder Years, ran with the concept and branched out like crazy, turning the show into a convoluted mystery built on Anachronistic Order and Continuity Porn.
The short-lived sitcom "Good News" is this to Amen as they were both created by the same people, took place in a church, shared the same sets, shared a Hettabrink sister (Amelia), and most of the plots involved an Amoral Attorney and the church's reverend. The only thing that's different on GN is that there's no Thelma or Rolly, and GN is set in inner-city Los Angeles while Amen is set in Philadelphia.
The Lifetime mini series, Marry Me is a spiritual successor to Maneater.
My Summer With Des (one-off slice-of-life dramedy by Arthur Smith against the backdrop of Euro '96) is a spiritual successor to An Evening With Gary Lineker (one-off slice-of-life dramedy by Smith and Chris England against the backdrop of the 1990 World Cup).
Jessie could be considered a successor to The Nanny, since they are both about regular women being hired as nannies for wealthy families through pure happenstance.
Not to mention that the creator/executive producer for the former, Pamela Eells O'Connell, was one of the original writers for the latter.
Homeland is a Spiritual Successor to 24. In addition to sharing a lot of writers, executive producers and production staff, both shows are concerned with intelligence and counter-terrorism work, what motivates terrorists and double agents, the personal costs of such a life (both terrorists and counter-terrorism agents) and the lengths that both sides will go to. Homeland, however, skips 24's major gimmick.
Downton Abbey can be considered a spiritual successor to Upstairs Downstairs, given the similar themes (both deal with the lives of a large aristocratic family and their servants, both are period pieces, and both feature numerous characters) that both shows share. As a bonus, Julian Fellowes, the creator of Downton, even admits to his show being a successor to Upstairs Downstairs.
Twin Peaks was a Spiritual Successor to an unrealised Mark Frost/David Lynch plan to dramatise the life of Marilyn Monroe. Both stories featured the mysterious death of a beautiful blonde with a murky secret life, all recounted in a secret diary. Lynch's film Blue Velvet could also be considered the spiritual ancestor of Twin Peaks.
"Reaper" to "The Loop". Both had Bret Harrison playing a guy named Sam, who hangs out with his slacker friends. And they both got worse the second season, although YMMV on The Loop. Also Bret Harrison played a guy named Sam on Law & Order: Special Victims Unit, who (like in Reaper) had a blonde mother named Linda.
Cougar Town to Scrubs. Both have the same creator and share many of the same writers and actors, and are very similar in tone and humor. They even share at least one character. One episode lampshaded it in the opening title. "Welcome to Cougartown. No, it's not just Scrubs in Florida with a lot of wine."
Community also feels like one. "Mundane" setting, every character is strange at the very least, lots of improbable Bunny Ears Lawyers at anything, Loads and Loads of Characters all known for a quirk or a gimmick (POP POP!), and a lot of "unique" episodes which do something different.
Tin Man has Spiritual Succesors in the form of Alice and Neverland
Bunheads to Gilmore Girls: both created by Amy Sherman-Palladino, both have a fast-talking, pop culture savvy, cynical (though Michelle is INFINITELY more bitter than Lorelai), thirty0something female lead. Both take place in small towns, both shows share Kelly Bishop, Chris Eigeman, and others, etc.
Battle Dome to American Gladiators. Michael O'Hearn was in both (as Michael O'Dell in the former and as Titan in the latter's revival). Also, Terry Crews' schtick as the Old Spice spokesman could be a Spiritual Successor to his T-Money character on Battle Dome.
Since both shows involve a protagonist played by Melissa Joan Hart who is a quirky Cool Loser with an interest in journalism and a rather unique first name, more than a few people have described Sabrina the Teenage Witch as "Clarissa gains magical powers."
The BBC's Atlantis is a spiritual successor to the much loved Merlin, drawing influences and ideas from classical mythology and history rather than Arthurian legends. The series is also built around the relationship between the male leads, although Jason, Pythagoras and Hercules are considerably less vitriolic than Merlin and Arthur.
The Goldbergs is essentially the 1980s set version of the 1960s set The Wonder Years, sans the warm fuzzy nostalgia. Both feature a family of five, with the main character being the youngest of three siblings, along with his aloof sister and pain in the ass older brother. Both feature a sometimes grouchy father with a subtle heart of gold, and both feature ongoing retrospective narration from the now-adult main character.