Sam Malone in Cheers. He went from earthbound jock to idiot after Diane left.
This was uneven. Some episodes he was still portrayed as being reasonably bright, others he was a lecher dominated entirely by his reproductive organs. It went back and forth depending on whether Sam was being a reality check for Rebecca or trying to bed her.
Jayne over the course of the short run of the Firefly series seems to go from unsophisticated but frighteningly cunning in the pilot to just plain stupid.
Cat from Victorious. She goes from being of relatively normal intelligence in the early parts of Season 1 to a girl who is apparently intelligent enough to just color a "pretty tiger" purple and realize that "that doesn't happen in nature".
Robbie also qualifies, as many of the problems the kids get into over several episodes (notably The Great Ping-Pong Scam, How Trina Got In and Locked Up!), are directly his fault. His behavior in dealing with Rex (notably in Wi-Fi In The Sky) also puts him in this category.
Michael Scott for a bit, but they reined it back in.
Inverted with Dwight who started the series as a gullible man with almost no common sense and then became more and more skilled and successful as time went on. By seasons six and especially seven he seems smarter than any of the other characters.
Kevin Malone. He was never the brightest bulb at Dunder Mifflin, but he showed signs during the first few seasons of just being a little dopey. A savant at the worst. By Season 5, he had lost about 25-50 IQ points. As of Season 8, he is pretty much mentally retarded.
Erin, who replaced Pam as receptionist. When she was introduced she started off as a person of average intelligence to not knowing how disposable cameras work the next season.
In fact, everyone on Heroes gets this, with nominal good guys Mohinder, Peter and especially Hiro getting the worst of it.
The villains too. Adam Monroe goes from being a coldly calculating bad ass with mad samurai skillz to a guy who meekly allows powerless brainless thug Knox to take him to see Arthur Petrelli. Arthur himself goes from an Evil Overlord big on the "You have outlived our usefulness" who was genre savvy enough to take out everyone who could oppose him to being a dumb ass who stands around and does nothing while Sylar has a bullet aimed at him. The German in the graphic novels was a vicious amoral killer who knew how to use his powers in clever and brutal ways but in the show he's an idiot who just does nothing while Knox lumbers over to him and slams his fist through his chest. Candice goes from being a smart and sadistic villainess to being stupid enough to let her guard down around Sylar. Elle has the exact same thing happen to her.
Initially he was quite intelligent and a prime example of Colonel Badass. He had tendencies of The Watson, but there's no shame in knowing less about science than the one who blew up a sun, less about ancient cultures than the one who's studied them all his life, or less about alien technology than the one who lived with it. Then the things being explained to him got simpler and more obvious as his jokiness quotient rises. At the nadir, we get "La, la, I can't heeeear you!" when finding out a guest star was in fact someone incredibly dangerous from a previous episode, or later gems like "SHE'S HAVING HER BRAINS SUCKED OUT!" By season five, his role seems to be "make jokes during dead-serious situations while others roll their eyes and make the decisions that would have been O'Neill's responsibility." An episode had the people of the planet of the week talking about how Earth would have sent its best and brightest representative... and then O'Neill enters, making one of the sort of inane comments that had become normal for him. When the idea of the leader of the team being a smart and capable guy is a punchline, you know you have a problem. Later seasons seemed to go back and forth with his intelligence.
Rimmer from Red Dwarf, though how many levels taken seemed to change with every season. The Cat is an even better example having started out as... a cat and ended up with "a brain the size of a grape."
Holli, the ship's A.I. went from being extremely eccentric after spending 3 million years alone and Obfuscating Stupidity to literally having a single digit IQ due to "computer senility" a couple of series later.
Joey from Friends seems to have lost quite a few IQ points over the years, to the point where in the final season he can't tell left from right.
Also Erica, the biological mother of Chandler and Monica's twins. When we first meet her she's a perfectly normal, intelligent girl. When she appears later in the season she's so stupid that her IQ is merely half of Forrest Gump's.
Chrissy on Three's Company was originally mildly ditzy. By the time she left the series one wondered how she managed to dress herself or hold a job.
Baldrick from Blackadder — there's the obvious leap between series one and two (while Blackadder himself does the opposite and takes a level in smartass), but even after that, he does still manage to get gradually worse over the next three series, until in Blackadder Goes Forth he's what Tony Robinson described as "terminally stupid" — you literally can't imagine how he's survived to adulthood. Of course, in Blackadder Goes Forth his childlike personality is both hilarious & tragically poignant, given the hopelessness of the World War I trench setting. He even gets some moments of childlike wisdom in the finale, simply questioning why they can't all just stop fighting and go home.
If you want concrete evidence, compare this scene from Blackadder II...
Edmund: That Farrow bloke you executed today, are you sure he's dead?
Baldrick: Well I chopped his head off, that usually does the trick.
...with this one from Blackadder The Third
Edmund: [...]So, what's the plan?
Baldrick: We do... nothing.
Edmund: Yep, that's another world-beater.
Baldrick: Wait, I haven't finished. We do nothing until our heads have actually been cut off...
Edmund: ...and then we spring into action?
The character changes between series are all justified because it is a different Baldrick in each one.
The same goes for Melchett and his descendant from Blackadder II to Blackadder Goes fourth.
Matthew Brock from Newsradio is another example; in the show's first season he seemed relatively normal and competent (he even secretly was a registered dentist), but by the third or fourth season he came across as either mentally retarded or an eight-year old in a man's body. Jimmy James to an extent too, although that was more of Taking a Level in Craziness than Dumbness. Both characters were a lot funnier and more interesting due to these changes, of course, due to the quality of the writing & acting.
While Matthew has no excuse, Jimmy James may have been engaging in Obfuscating Stupidity. Alternatively, he really is that eccentric, and his business skills are only used as he needs them. This is apparent given that Mr. James is implied to be a highly successful Self-Made Man, and his occasional bouts of serious behavior back up that he didn't get there by being an idiot. Most likely, he really is a goof ball and we generally only see him when he's relaxing, rather than engaging in Serious Business (which is on several occasions implied to involve questionably legal activity).
Radar O'Reilly of Mash could be regarded as a variation of this. He didn't get dumber, per se, but more childlike and na´ve as the series progressed. There's an element of Ping Pong Na´vetÚ there too, however, as he was apparently still savvy enough to make deals for supplies with other units, keep all the paperwork straight, and otherwise serve as Hypercompetent Sidekick to Colonel Blake (and, later, Colonel Potter).
Possibly justified. A recurring theme of the series was that people found odd and near-insane ways to cope with what they were experiencing. Radar's steady retreat towards childhood is actually one psychological response people have to stress in the real world.
Radar's character regressed in a lot of ways. Early in the series, he was seen smoking cigars, drinking Henry's booze, and had several references to him peeping in the nurses' showers and enjoying developing their chest x-rays. As the show progressed, he got dumber, more naive, and less experienced to the point that even grape Nehi was a strong drink for him.
Nate Archibald on Gossip Girl. While he wasn't as intelligent as Chuck or Blair in season one he was still a guy of normal intelligence. By season four even his actor is complaining about what a moron he has become.
In fairness, he's done so much pot in the last four seasons it actually does make sense that he's dumber.
Granted, Potsie Webber of Happy Days started out a little on the dim side, but he only got dumber and dumber and dumber as the seasons went on.
Nearly everyone from Charmed but Paige was probably the worst affected, going from one of the best things about Seasons Four and Five to becoming a spazzy idiot. Best example of this? ORB THE FUCKING GUN, RAIGE!
Charlie on It's Always Sunny in Philadelphia. He was always the least intelligent/educated member of the group, but over the course of the first few seasons he got hit with this trope so hard he lost the ability to read and write (though he seems to think he can).
Mac: Dammit, Charlie, your illiteracy has screwed us again!
The title character from Everybody Loves Raymond was flanderized from being a witty, intelligent Every Man who wins national writing awards into becoming a bumbling buffoon over the course of the show, so that the creators could more easily shill his wife as "the reasonable one", pander to her fanbase for ratings, and make her the DesignatedHeroine of the show in the later seasons.
KC from Degrassi, he's introduced as a member of the school's gifted program, two seasons later he thinks there's four trimesters during pregnancy.
Randy Disher on Monk seems to get more and more improbably stupid as the series progresses.
This is even lampshaded in one of the novels. the Captain explains while he is an idiot in homicide cases, he is very good at the smaller crime cases.
Monk himself isn't immune to this trope. As the series went on he seemed to morph from a brilliant but troubled detective to little more than a quivering mass of OCD hangups who just happens to work for the police on cryptic crimes.
Robin Scherbatsky on How I Met Your Mother didn't get dumber per se, but she did acquire a lot more quirky traits, gaps in her understanding of American culture, and general insanity (*salutes*) in late season 1 through early season 2. The relative mildness of this effect meant it served to make her character a lot funnier and well-rounded: before, she had been an interesting character, but had a fairly narrow, undeveloped scope. This trope also actually made sense here, as she wasn't really a part of the gang in season 1, when her connection to the other characters was being Lily's new friend and the girl who Ted had a crush on, so she wasn't especially open with them and didn't spend as much time with them as she would later on.
Jake on Two and a Half Men transformed from a naive, lazy kid into a brain dead pothead in later episodes.
Reese in Malcolm in the Middle always had below average intelligence, but as time went on, it went from failing school to his homework being mistaken for something Jamie made. (That said, it was also occasionally implied that he was actually as smart as Malcolm, but just didn't apply himself.)
Dr. Drew Baird, Jon Hamm's character on 30 Rock. Status Quo Is God, so something had to end his relationship with Liz, so we got "The Bubble," in which Liz realizes that Drew is a barely functional idiot and doesn't know it, having been happy and successful his whole life because he's so handsome that people will do anything for him. This is an especially egregious example because his characterization up to that point had been "ridiculously perfect, especially for Liz," and because his two previous appearances had provided equally plausible reasons for a break-up if one was needed (Liz's stalker behavior, and the Plethora of Mistakes that was their first attempt at a date).