Keiko Takemiya's Kaze to Ki no Uta is the spiritual successor to "The Door To Summer", a one-volume manga she wrote. They both contain similar story elements, as well as both being adapted into 60-minute OVAs.
Both of these series in turn are spiritual successors to In the Sunroom, a lesser known manga of hers, which also have similar story elements of the two series mentioned above.
Gokukoku No Brynhildr can be considered a successor to Elfen Lied, seeing as they are both written by Lynn Okamoto, contain cute supernaturally empowered girls who are mistreated in the name of science, gory fights and have harem undertones.
The Big O could be seen as a successor to Daitarn3, only using Batman instead of James Bond.
The Big O itself has a spiritual successor in Tiger & Bunny. Both are anime series by the same studio, share part of the staff, and has the look and feel of an american saturday morning cartoon. While the first sported a mecha pilot Bruce Wayne, the second depicts the results of a government/private take-over on the Justice League. Also including a cute short-haired android voiced by Akiko Yajima.
Ninku shares many similarities with Naruto despite coming first. Examples like the fact that Fuusuke is a wind ninja who wears orange and uses the kuatsuken (resembling Rasengan).
Fairy Tail is this to Mashima's earlier series Rave Master, containing a few characters with the same name and/or design, and even a few plot elements with the same name but different functions. Eitherion, Oracion Sei, etc.
It's sometimes considered one to One Piece as well.
Durarara!! is this to Baccano!, although the two are different in tone and setting, they share the same creator, the same "Loads and Loads of Characters with distinct plotlines on a collision course" writing style, and the same jazzy musical style.
They also happen to share the universe, though taking place in different countries and decades apart, only two characters from the first make a brief cameo in the second.
Many fans are also calling Mawaru-Penguindrum successor to Utena; again, same people working on it.
Tamayura is the spiritual successor to Kamichu!. Not only is the show about a tightly-knit group of teenage girls, with the mood, art style and theme practically lifted from its illustrious predecessor, but it also takes place in Takehara, an old town not far from Kamichu!'s Onomichi and similar to Onomichi in many respects. Still, an argument could me made for Sketchbook as an influence, considering the age of the characters and the art theme (not to mention the opening).
Nodame Cantabile is seen as the spiritual succesor to Honey and Clover. Both are about students in art schools (drama ensues, obviously), both anime adaptations were made by the same studio, with a very similar drawing style.
Spotted Flower is one to Genshiken, both by the same author and focusing on the otaku culture, although there are subtle hints pointing that the former might be a true Sequel to the later, with the two of the main characters grown-up and married.
Guilty Crown: General consensus is that this series was a partial one to Code Geass, particularly in the first few episodes, given that it shared the same "resistance group taking on oppressive entity" premise. While Shu and Lelouch have very different personalities and interests, they are both 17-year old youths accompanied by mysterious girls (Inori and C.C. respectively, who also differ from each other) and capable of using mysterious powers. However, the overarching story eventually headed in more of a different direction.
Dusk Maiden Of Amnesia can be considered this to Natsu no Arashi! in animated form at least - both series were directed by Shin Oonuma and feature cute (actually, near-identical) Ghost Girls and a healthy dose of SHAFT-like visualsnote though only Arashi is a SHAFT production itself - Dusk Maiden is produced by Oonuma's spin-off studio, SILVER LINK.
In addition to having the same creative team, both Kill La Kill and Gurren Lagann share similar stylistic elements and cover similar themes. Where the latter is about boys growing up and learning to face the challenges of life (told though over the top Super Robot action), the former is about girls growing up and learning to grow past the expectations and pressures of society by way of Magical Girl Warriors wearing sentient school uniforms.
Kamisama Kiss has a lot in common with InuYasha. To begin with both plots revolve around an ordinary teenage girl getting control over a supernatural asshole, who is forced against his will to help her and protect her, and only later do the two start to fall in love. Tomoe, the male lead in KK, looks almost exactly like an older and more mature version of Inuyasha when he wears his hair long; he also acts like a more intelligent and slightly more mature version of Inuyasha. Nanami, the female lead in KK, looks like Kagome with brown hair and has the same personality type. Then the Time Travel Arc happens in KK and the similarities get even more pronounced.
Captain Earth towards Star Driver. It helps that both series feature the same staff of people (except for character design) and concepts from the latter show are used again in the former, most prominently the usage and explanation of the term "libido". Since Captain Earth is still running, the jury is still out wether both series are set in the same universe or not and speculations are running rampant. What can be said already, however, is that Captain Earth has an overall more serious tone than Star Driver did.