Obsolete Mentor

Basically this is someone who was a master in their field, but they're now behind the times as they haven't been able to keep up with advances in the field. Particularly using newer technologies. This can be done in three ways:

They become a mentor to the hero, teaching him ancient and powerful techniques to supplement (or replace) his arsenal.
They allow for An Aesop on not getting over-confident or complacent, by dying or getting Put on a Bus when the modern techniques they needed could have saved them.
They still remain awesome, despite (or because) of forsaking the advances. Sometimes the old method is supplanted by an apparently-more-powerful new method, but the new method has a terrible curse that leads to its ultimate defeat or rejection.

This is, of course, Truth in Television. Professionals in many fields must constantly undergo training to keep up with the state-of-the-art. Could result in or be a result of Surpassed the Teacher.

Examples of Obsolete Mentor include:

Anime and Manga

Master Roshi is a prime example of this. At the first World Martial Arts tournament (the second major storyline), he knew that would happen at the rate Goku was going at. By the first saga of Dragonball Z he had become absolutely worthless, while most everyone else had a least another season or two before that set in.
- Lampshaded by himself on the Cell arc, where seeing how everyone else can at least try to help while he just watches, he muses about the time he was "Muten Roushi, the strongest man in the world"
- King Kai also becomes this after the Saiyan Saga, though he's still a good source of information.
One of the themes in Naruto is that the younger generation will eventually surpass the older one and will also protect the even younger and unborn generations so that they can take things over in the future. A good example is Naruto himself. He's taken the Sage training of his mentor Jiraiya along with the Rasengan created by the Fourth Hokage and enhanced them using a specialized training method involving Shadow Clones, a jutsu we later learn his mother was fond of.

Film

In Big Fat Liar, the stunt-coordinator for the movie was one of the first and best in his time, but now he's out of a job. ("That stunt looked impressive when I first saw it IN 1945!!!") This actually turns out to be a Chekhov's Skill.
Robert Duvall was the mentor in Gone in Sixty Seconds. He was still effective in his field, but had to contend with younger, more hi-tech thieves in the crew. One of them showed off his fake fingerprints: the mentor still preferred gloves.

Literature

To Crabbe and Goyle from Harry Potter, Draco became this once the pair of them discovered their unique penchant for ruthless torture, and the Malfoy's name became mud with The Dark Lord.
In one of the arguably nonexistent prequel series to Dune, the School of Sword Masters exemplifies dedication to physical skill, spiritual balance, and a code of honor. At the end of a main character's training with them, their school comes under surprise bombardment and invasion by a mercenary force avenging a student they'd rejected on moral grounds, and is rapidly destroyed. Despite the masters (at least mostly) surviving, and vowing to rebuild and correct the mistakes which led to this, they do drop out of the story at that point.
Due to having been born in 1900 DCI Nightingale from Rivers of London sometimes falls into this trope when it comes to modern police-work and technology. When it comes magic though, well this is a man who personally blew up two attacking Tiger Tanks in WWII on his own.

Live Action TV

An episode of Scrubs had Kelso forced to fire his best friend as he couldn't keep up with advances in medicine.
An episode of Murphy Brown revolved around Murphy's sadness at seeing her former mentor become this (for example, despite the show being set way into the 90's, said mentor believes telex machines are still used).
On Star Trek: The Next Generation, Scotty had been stuck in a transporter loop for over 70 years. He was able to help Geordi with their current adventure despite being decades out of date.
- Of course, Scotty literally wrote the book on all modern-Starfleet engineering, but freely admits to padding his estimates and withholding surplus power to look good. Basically, the reason Scotty is still a master is because modern Starfleet still haven't figured out that by following Scotty's bible, their starships are actually kept from being half as powerful as they should be.
On 30 Rock, Liz received mentoring by a woman who wrote television in The Seventies (played by Carrie Fisher). Her bold, boundary-pushing writing style was too much for the modern, corporate-controlled NBC. Liz ultimately abandoned her after seeing how her life had ended up.
Corporal Jones from Dads Army: Respected as a soldier who served under Lord Kitchener in the Sudan, but now a confused old fool.
There was an Outer Limits episode about a librarian who couldn't connect to the mind-linked Internet of the future and was looked down on for actually reading books. Of course, when the network went haywire, he was the only one who could help.

Music

The folk song "John Henry" is all about the technological obsolescence version of this.

Newspaper Comics

Parodied in Dilbert: Bob the Dinosaur gets hired in the Y2k run-up as a COBOL programmer because he says he's told he looks like one.

Video Games

Trauma Center: Under the Knife 2 introduces the Hands of Asclepius, a rival medical institution to Caduceus which used a combination of Derek Stiles' performance data and performance-enhancing drugs to mass-produce surgeons armed with Derek's Healing Touch. After Derek is unable to save protege Adel Tulba during his Heroic BSOD, leaving his operation to someone else, Adel resigns from Caduceus and joins the HOA, becoming one of their unnaturally-empowered super surgeons. As the game proceeds, Caduceus is pushed further and further into the background as the HOA gains prominence; however, it's soon discovered that their progress comes with a terrible price: the performance-enhancing drugs are actually a new strain of the deadly GUILT, and every single one of HOA's super surgeons becomes a half-crazed walking time bomb.
Kiritsugu in Fate/Zero is able to overcome some very tough magi and in fact assassinates them for a living. Why? Because, in the words of magi themselves, science goes towards the future and magecraft towards the past, so they shun technology and the advantages they can give.
Similar to the above and equally not-in-the-first-category of actually being a mentor, Caster in Fate Stay Night is far better at magic than Tohsaka... but she hasn't realized that because magical aptitude has fallen somewhat, Magi have taken up physical defense as well. She is therefore is therefore beaten to a pulp when Tohsaka closes the distance between them.
- This is zig-zagged a bit during Fate/hollow ataraxia, however, with the presence of Bazette and the Fragarachs. Though Bazette is a very gifted combat mage, Nasu still stated that Caster would cream her in a fight, since Fragarachs may be an instant kill attack, but can only be used to counter an opponent's "best attack", and Caster is too well-rounded when it comes to spellcating to make her vulnerable to it.

Web Comics

In Eight Bit Theater, Fighter is warned to stop taking his skills for granted to avoid this trope. He deals with this advice in his usual manner.
Specifically stated in an early chapter of Girl Genius; when an army of Beetle's clanks (once the best in the world) are dispatched in a single panel, Klaus gives him a lecture that essentially embodies this trope.

Real Life

According to legend, when Leonardo Da Vinci (as an apprentice) was finally allowed to paint part of one of his master's works (in those days artists would have assistants and apprentices do at least some of the work for them) his master was so impressed by the result he gave up painting on the spot.