Tropes Are Tools, but some have aged better than others.
Over the course of time, a trope may be overused, misused, opposed, made obsolete, subverted on many notable occasions, or just end up being widely disliked. Eventually, a trope may reach the point where it becomes one which nobody should dare use seriously and only belongs in parody, satire, homage or pastiche. Often, if one of these is used straight, people will assume it's a Red Herring.
In some cases, a trope may be discredited due to changes in our knowledge of history or science. Use of the trope in fiction may change to reflect this. See the Time Marches On index.
Omnipresent Tropes are immune to being discredited, mostly because those tropes are too natural to the medium of storytelling to ever be considered tired cliches. Undead Horse Trope describes tropes that have been subverted and parodied dozens of times, but aren't quite discredited.
Dead Horse Trope, where subversions or parodies outnumber straight use in recent works.
Forgotten Trope, which describes tropes that aren't used in recent works at all; they may have been considered Discredited Tropes years ago, or just fell from use for other reasons.
Awful Wedded Life: Back in the day, there was much more stigma against getting divorced, and couples were expected to resolve their problems themselves or simply pretend marital problems didn't exist. These days, however, at least in the Western world, divorce has lost much of its stigma, divorce laws are more liberal (making it relatively easier for couples to get a divorce) and it's much more commonplace than it once was, so far fewer people these days are trapped in a loveless and unhappy marriage.
Barbaric Bully: Heightened school security in the post-Columbine era means that beating a kid up in a crowded school hallway usually comes with consequences. Not to mention that the advent of cyberbullying and the recent rash of bully-related suicides proves that a good deal of bullying is psychological rather than physical (and that psychological bullying can be just as harmful as physical bullying). Not discredited in British works due to Values Dissonance.
The Bermuda Triangle: No matter what anyone wants to say about Real Life ships and planes that have disappeared in the region, aliens, Cthulhu, dimensional portals, Atlantis, etc. are NOT behind any of it. The incidence of Real Life disappearances in the area is no higher or lower than any other part of the ocean that has similar size, weather, and maritime traffic.
Bedsheet Ghost: A hopelessly outdated cliche of a ghost appearance that is very difficult to play straight now.
British Royal Guards: Never used for anything other than comedic effect, but nowadays the once common gags involving a guard's effort to remain still under immense pressure have been replaced with ones where voluntary movement on the guard's part is observed, side-stepping more commonplace expectations.
Cement Shoes: It's much more efficient (from both a filming and story perspective) to have the crooks just shoot the guy or bash his head in and go on their way. The victim being suddenly shot also has more shock value.
Chocolate-Frosted Sugar Bombs: Modern children's cereals are made with much more of an eye towards nutrition, thanks to backlash regarding marketing junk food to kids.
Christmas Cake: Along with its Western variant, the Old Maid, it's not quite gone from consciousness, but it is on its way out, thanks to changing attitudes about the role of women in society, and increased education for girls.
Cut-and-Paste Note: In modern fiction, due to the prevalence of more convenient and harder to trace forms of anonymous communication. If used in any sort of forensic drama, you can bet the CSIs will admonish the culprit as an amateur and get damning evidence off the note.
Cut Phone Lines: Largely discredited in any story set after the widespread adoption of cell/mobile phones.
Declarative Finger: Often used by the authors to imply that the character doing so is just trying to come across as profound, which in turn is used to imply that the character is actually saying something NON-profound.
You see this every day, so it's not really a plot device.
There's also the common variant of the trope, "Does this make my butt look big?". Because of evolving beauty standards in the 21st century, the implication that a woman has a big butt is a lot less likely to be considered an insult than it once was.
Girls Have Cooties: Unless you are aiming at a VERY young audience, using this trope will make you a laughingstock.
Henpecked Husband: As Many Domestic Abuse cases start to have more gender neutrality be added to them, this type of relationship is seen as either unrealistic or horrifying. Many sitcoms still use them, however, as do animé, manga and similar.
Mayan Doomsday: December 21, 2012 is in the past now, and suffice to say, the world and civilization in general remain intact. It's hard to imagine any new works taking this trope seriously with that in mind.
The Natives Are Restless: This isn't the age of colonialism anymore. It still pops up in fantasy and science fiction settings where colonialism can exist, but even then, it's more likely to be used for humour.
Not Allowed to Grow Up: Still largely used and accepted in animation, comics and video games, but discredited in live action media.
Phone-Trace Race: Still used on occasion by very dense Hollywood hacks, but with caller ID, the popularity of shows like 24 which have mostly ditched this trope, and a general paranoia about Google and Facebook tracking your every move, writers nowadays tend to err on the side of the FBI/NSA/CIA being too good at tracking your every move.
Santa Claus Tropes: Santa has become such a commercial icon of Christmas and as such overexposed via countless Christmas specials and merchandise, that it is pretty much impossible to play any trope related to him straight now, unless you have a really young audience in mind or have no self-respect for yourself as a storyteller.
Shotgun Wedding: A combination of greater availability of contraception and a reduced stigma of single motherhood make this trope seem very dated today. If a woman does get pregnant, a couple doesn't have to get married, but the father is expected to provide some form of support.
Standing in the Hall: Parodied in some Japanese works still; but not used in Real Life as much. In western countries, similar variants aren't used due to kids taking it as an opportunity to wander around the halls.
Stern Nun: For one thing, many teachers at modern Catholic schools are laypeople, not nuns. For another thing, after the advent of the Self-Esteem Movement of The Seventies, parents would be all up in arms if a teacher (whether a nun or a layperson) meted out those kinds of punishments. (Nuns also haven't worn those starchy white wimples since the 1970s at the latest, but you'll still occasionally see them, usually for humor.)
Stranger Danger: Aesops about the inherent dangers of talking to strangers were huge in the 1990s, but due to changing attitudes (coupled with the discovery that the majority of abuse comes from people that the victims know personally) it's rarely played straight.
The Generation Gap has pretty much become this, with Generations X and Y becoming parents and today's children growing up in an unusually similar cultural landscape to the one their parents grew up in. Fortunately, recent shows like Modern Family and Two and a Half Men have realized this, and thus, the trope has been gradually phased out of television and movies during the past decade-and-a-half or so. Today, it's relegated mostly to parody, although even that is becoming old hat.
On that note, Give Geeks a Chance has mostly become this. Now that the trope has become an overdone cliché and the mainstreaming of nerd culture has rendered it more or less obsolete, it's very rare to see it played straight nowadays.
Alliterative Name: A trope popularized by Stan Lee so that he could remember all these characters he co-created. It's been taken to the point of parody.
Superheroes Wear Capes: Time was when every other superhero had a cape. These days, capes are considered to be impractical, and even dangerous (see the fate of Dollar Bill in Watchmen or Edna Mode's speech about cape-related mishaps in The Incredibles). You'll still see capes on plenty of DC heroes such as the Superman family and the Bat family, but mostly because of the Grandfather Clause. Aside from Doctor Strange and the supervillains Magneto and Dr. Doom, Marvel has very few characters who wear capes.
Said Bookism: In these days, it's often considered redundant.
Prolonged Prologue: Often seen in fantasy novels. They give a lot of backstory on a world that the reader hasn't read enough of to care about yet. Most publishers say Show, Don't Tell and let the reader learn about the world through the eyes of the characters.
Truck Driver's Gear Change: It's become such a cliche, especially during the second half of the twentieth century, that it's now almost impossible to play it completely straight anymore.
Digital Piracy Is Evil: Despite still lingering today, companies have ultimately realized that the war against piracy is a lost cause, and have taken incentive to work around it instead. More recently they have been pushing a new bill (s.978, Protect IP, SOPA) to put an end to piracy forever, although all attempts so far have failed. Although in the United Kingdom, the Digital Economy Bill may keep this as not quite a discredited trope, as it seems that public opinion is against the bill, despite politicians' attempts at copyright law changes. Values Dissonance, indeed.
That Reminds Me of a Song: Modern musicals, at least in theatre, are specifically not supposed to play this one straight anymore, though there's still a chance a song of this nature may end up as a Breakaway Pop Hit
"I Want" Song: This became discredited for a while after Disney and its competitors milked the Broadway musical cartoon formula for all it was worth — the makers of Toy Story even intentionally avoided this, in order to distinguish it from those films. That said, there's enough nostalgia left for it now to allow it to return in recent films like The Princess and the Frog, but it's nowhere near as prevalent as it was in the past.
Rotoscoping: Derided as a lazy, poor substitute for actual animation by both animators and critics (thanks in part to another trope it inevitably invokes), it is almost never used in modern hand-drawn animation, let alone without irony. It's more modern equivalent, Motion Capture, is still used in CGI, but often to equal derision.