Tropes Are Not Bad. But some tropes haven't aged well.
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 none 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.
Note: Just because a trope is discredited does not necessarily mean it is not Truth in Television.
Note #2: This is not Bad Writing because the writing itself is bad, but because the writer doesn't know its audience. After all, Tropes Are Not Bad.
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.
See also Dead Horse Trope, where subversions or parodies outnumber straight use in recent works. See also 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.
Compare Discredited Meme.
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.
Celibate Hero: Not taken seriously anymore in fiction unless the hero is simply physically or medically unable to have sex. Even then, romance at least is assumed.
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 of the note.
Cut Phone Lines: Largely discredited in any story set after the widespread adoption of cell/mobile phones.
This has caused the trope to generally morph in various ways: the area has no reception, dead/missing battery, interference due to solar flares, etc.
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.
Face on a Milk Carton: Thanks to, in the U.S. at least, the Amber Alert system which allows missing children's names to be broadcast on television or on expressway signs within minutes.
Officer O'Hara: There are still Irish-American cops in entertainment, but they tend to be less stereotypical. The whimsy and the just-off-the-boat accent tend to only be used straight in Historical Fiction these days.
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.
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.
One contemporary airplane disaster movie tried to play itself straight for the first part of its production process - then someone realized there were motherfucking snakes on the motherfucking plane.
Literature
Said Bookism: In these days, it's often considered redundant.
Live Action Television
The line "Hi, honey, I'm home!" was a stock standard phrase in many American family sitcoms from the 1950s and 1960s. Back then it was used straight forward, but since then it has been discredited due to its corniness and unrealistic routine.
New Media
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 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.
Screamers have received two major blows over the Internet's history. Initially, when flash movies and games were still the norm, there were no clear distinctions between screamers and legitimate pages, creating a minefield for fearful site goers; this meant less traffic for sites like FunnyJunk and WinterWorld. Later, with the advent of video over flash files, viewers were able to scroll to the end of the video to see if any suspicions were confirmed, removing all suspense and defeating the purpose of screamers. They have since been replaced by the trap video, which puts the scare at the beginning of the video, and aims not to make individuals jump, but to cause outrage within specific audiences.
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
Monster Closet: In first-person shooters. Present in shooters in mid 1990s to early 2000s but mainly replaced by offscreen or onscreen spawning.
One Bullet at a Time: Subjective; was originally a technical limitation, but can still be enforced for gameplay reasons (i.e. prevent some forms of Spam Attack).
Random Encounters: As a remnant of technical limitations of video games and its tabletop origins, they're lately replaced by other methods to engage a fight.
Some games made in RPG Maker play with this trope, by having the "Random Encounters" actually be regular encounters, but with the wandering monsters being invisible.
Tabletop Games still use random encounters fairly frequently, where they're just a trope used in some games where they fit the flavor better. Additionally, some pretty big games such as Fallout 3, many MM Os, any dungeon-crawler patterning itself after Diablo, and most of the JRPG genre still use these types encounters. This may be an Undead Horse Trope instead.
Video Games Are For Nerds: This was gradually becoming discredited when the Playstation 1 was released. By the time the PS2 became popular, it was pretty much dead. Yet, many gamers (probably as a symbol of pride) still seem to hang on to it.
While still common enough the target seems to have moved a bit with the nerds now only being obsessive or interested in a particular genre (e.g. MMORPGs) or niche (e.g. Japanese dating sims). On the other side there has also been a rise in Video Games are For Frat Boys, again, depending heavily on the games being depicted (FP Ss and sports games seem to be the most common) and their attitude towards them.
Western Animation
"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.
This trope is not as discredited in Western Animation these days as it was in Golden Age western animation though.
Not entirely discredited, but at least diminished since the 1990s are the fastpaced "cartoony" cartoons with gimmicky sound effects, weird body transformations and chase scenes. A lot of cartoons nowadays have more realistic action on the pace of The Simpsons, which resembles live-action TV more closely.