The Beverly Hillbillies ("...So they loaded up the truck and moved to Beverly. Hills, that is. Swimmin' pools, movies stars...")
Interestingly enough, though, the story recounted in the theme song is actually incorrect; it states that Jed found oil while shooting at food, while the first episode of the series shows us that Jed always knew there was oil on his land, he just didn't know it was worth any money. He was actually willing to pay the man from the oil company to take the oil away for him.
Paul Henning, who created The Beverly Hillbillies, also used expository theme songs for the other CBS rural comedies he produced in the mid-1960s. To wit:
"Come ride the little train that is rolling down the tracks to the junction/Petticoat Junction..."
"Green Acres is the place to be/Farm livin' is the life for me!" ... "Dahlink, I love you, but give me Park Avenue!"
For a hilarious parody, search YouTube for "fresh prince gangsta version".
It's not just an online thing. Quoting a bit of the Fresh Prince theme to a friend in a crowded bar where the majority of patrons were tweens and teens in The Nineties quite often results in a Crowd Song situation.
Gilligan's Island ("...Five passengers set sail that day / For a three-hour tour / A three-hour tour...")
The theme actually helped to get the show on the air in the first place! Sherwood Schwartz had trouble selling it to CBS because the head of the network thought it would be a pain explaining to everyone every week why they were on the island, so he wrote an opening song to get the point across. It worked, but because it was a calypso and the show was set in the Pacific, it was drastically changed for the series...
The Nanny, in a deliberate retro spoof of 1960s shows ("... That's how she became The Nanny...")
The theme song to The Weird Al Show tells the long and convoluted story of how a very strange man came to get his own TV series ("Oh-h-h-h / This is a story 'bout a guy named Al / Who lived in the sewers with his hamster pal...").
H.R. Pufnstuf ("But the boat belonged to a kooky old witch/Who had in mind the flute to snitch...")
They've got a power and a force that you've never seen before They've got the ability to morph and to even up the score No one will ever take them down The power lies on their si-i-i-i-i-i-iiiide...
Power Rangers actually waffles back and forth with this quite a bit. Disregarding generalizations like "We'll protect the world," quite a few of them aren't (Dino Thunder, SPD, RPM) while a few (Zeo, Lost Galaxy, Operation Overdrive) do it quite heavily, with Operation Overdrive doing so GRATINGLY. Jungle Fury does this to such an extent as to render watching the actual show moot, as it basically gives everything away save the ending.
The mostly dramatic (and certainly not a sitcom) Sea Change manages a pretty expository theme tune: "Don't want to live in the city/My friends tell me I'm changing". Complete with shots of the drive out of the city to the coast.
The end of the Civil War was near when quite accidentally, a hero who sneezed abruptly seized retreat and reversed it to victory...
The first two seasons of Red Dwarf do this after the theme, which was disguised as a distress call describing the situation.
The closing theme was originally meant to be an Expository Theme Tune referring to Lister's overall plan to settle on Fiji when they returned to Earth ("I want to lie, shipwrecked and comatose/Drinking fresh mango juice"). Since this was never mentioned after the first few episodes it's largely meaningless, something the composer is quite happy with.
The theme to the McLean Stevenson sitcom Hello Larry is a hilariously clumsy attempt at this trope. It almost sounds like the producers just walked into the recording studio and handed the singer a copy of the pitch they gave the network, then told him to improvise a melody, change everything to second person and toss in some rhymes here and there. Not to mention the wonderfully narmish line "Portland is a long way from LA."
Here Come the Double Deckers
Parodied in The Misadventures of Sheriff Lobo whose theme song raves on about how brave and selfless the sheriff is (though we know he's actually greedy and incompetent) played over a montage of the disasters Lobo and his men cause. It ends with Sheriff Lobo, who's been striding down a line of deputies standing by their patrol cars, calling "Move 'em out!" whereupon they all drive off at the same time causing one big pile-up.
Two and a Half Men played with this trope a little. Charlie was assigned to compose a theme song for the animated adaptation of a comic book called "Oshikuro, the Demon Samurai". But Charlie was only given a very vague description of the show, making his lyrics narrating an inaccurate version of the Samurai's characterization (besides sounding happy and campy like a theme song from a 1980s cartoon). The entire episode shows Jake trying to teach his uncle about the actual story of the comic book and how he should write the song. In the end, Charlie sent the new version of the song (which the viewers never hear), but the studio preferred the first happier and inaccurate version.
Amusingly enough, Jake is furious with how horrible the campy intro is, but both Charlie and Alan enjoy it.
The Smothers Brothers Show ("My brother dear was lost at sea / Without his water wings / So now he is an angel / And he does the most amazing things!")
Space Cases ("Once upon a time at a school in outer space / There was a class of misfit kids from all around the place...")
Mister Ed ("A horse is a horse, of course, of course...")
Shameless may fit here, having one of the lead characters speak (as opposed to sing) over a basic tune, a la My Name Is Earl
Most 1960s to late 1980s tokusatsu had these types of themes to various levels. Some such as Fireman's OP only made smaller references such as to his henshin device, but others like most early Ultra themes hit you over the head with references to the show.
The one season Bruce Campbell wonder, Jack of All Trades, was almost a throwback: "In 1801, the Revolution had been won, and Uncle Sam's favorite son, had a job he needed done..."
One Foot in the Grave. The opening and closing themes sung by Eric Idle describe the main character, Victor Meldrew, quite well. ("It's true that my body has seen better days / But give me half a chance and I can still misbehave")
Parodied by It's Garry Shandling's Show ("This is the theme to Garry's Show/The theme to Garry's show/Garry called me up and asked if I would write his theme song/I'm almost halfway finished/How do you like it so far/How do you like the theme to Garry's Show/This is the theme to Garry's Show/The opening theme to Garry's show/This is the music that you hear as you watch the credits/We're almost to the part of where I start to whistle/Then we'll watch "It's Garry Shandling's Show"/This was the theme to Garry Shandling's show.")
That's So Raven ("It's the future I can see / It's so mysterious to me...")
Hannah Montana: "Who would have thought that a girl like me / would double as a superstar?"
The Goodies had two sets of lyrics over the course of its run. The first set listed some of the services the Goodies might be able to provide (A circus or a seaside pier/A sausage or a can of beer,/A stripper or a clown/Prices going down/We can make it happen here). The second was a more generic description of the Goodies themselves (Here we come, into town/Gettin' up, fallin' down).
Welcome to LazyTown / A place where you'll wanna stay You'll meet Robbie with his Rotten plan / And Sportacus saving the day Stephanie is new in town / And soon she and Ziggy are friends...
Wonder Woman, Wonder Woman./ All the world's waiting for you,/ and the power you possess./ In your satin tights,/fighting for your rights/And the old Red, White, and Blue!
Sonny With A Chance's theme song starts off with "Off to the races; I'm going places; Might be a long shot; Not gonna waste it; This is the big break; And it's callin' my name"
All but one man died There at Bitter Creek And they say he ran away! Branded! Scorned as the one who ran. What do you do when you're branded, and you know you're a man? Wherever you go, for the rest of your life You must prooooove you're a man.''
Wake up Dick and Dom, and get out of bed Get youselves dressed, there's a crazy day ahead. There's so many things to do and lots of people to meet In Da Bungalow and on the street... (Go Go Dick and Dom!)
The short-lived 1981 sitcom Open All Night took this to its logical extreme by having a theme song that told the entire life story of the protagonist, from birth to the present day.
This is the story of Gordon Feester
Born in Ohio the day before Easter
Had a normal childhood, did OK in school
Graduated from Columbus High in 1962…
In a rare modern example, The Exes has one as of it's second season. Previously the opening was just a title card featuring the leads, now it details exactly how they each got divorced, introduced by their divorce lawyer (who lives across the hall), and moved in together, "now just call us the Exes!"
The Slammer ("You've been convicted of a howling showbiz crime...")