In Raising Arizona, when H.I. runs into the supermarket from the cops, a muzak version of the movie's theme song is playing.
Happens once a movie in the Spider-Man Trilogy. In the first one, it's during the montage wherein New York gets to know Spider-Man — a vaguely stoner-ish guy plays guitar on the subway, singing a song whose notes are those of the original Spider-Man TV cartoon, though the lyrics are quite a bit more ridiculous. In Spider-Man 2, a busker is twice shown plucking a fiddle and singing the cartoon theme; the movie uses her as a bit of a Greek Chorus whose words haunt Peter. In Spider-Man 3, a parade in Spidey's honor includes the theme performed by a full brass band. One wonders who wrote it in-world and why... Incidentally, Spider-Man 3 is the only one of the three not to have it appearing at the end of the closing credits (the first film has the actual theme from the TV cartoon, the second has a rendition by Michael Bublé).
In Licence to Kill, Sanchez fires some bullets at Bond and misses. The sound of the bullets ricocheting off the tanker is set to the Bond theme. This apparently was a Easter Egg on the part of the sound editor.
In On Her Majesty's Secret Service, there is one scene where Bond walks past a cleaner, who starts whistling the Goldfinger theme, and during the "gadget scene", some old Bond themes can be heard; OHMSS theme can be listened often in the film, such as during the snow chase towards the end. The Bond series seemed to be quite fond of this trope.
In Live and Let Die, a New Orleans lounge singer sings a Motown-style reprise of the title theme.
Taken to its logical extreme in the movie The Long Goodbye where with the exception of the opening and closing of the movie, the only song heard in the movie is various arrangements of the theme song, "The Long Goodbye" used diegetically. So when a character turns on the radio, that's the song that plays, when a character is at a bar there's a piano player singing that song, in the supermarket, a muzak version is playing on the overhead, and when the nudist, hippie, neighbors, are chanting, they're chanting the theme also.
In Guillermo Del Toro's Pan's Labyrinth, Ofelia asks Mercedes to sing her a lullaby. Mercedes hums a gentle, mourning tune, which just so happens to be the movie's theme. This lends to the film's complex affair with the boundaries of reality and fantasy.
In the live-action Inspector Gadget movie, the theme tune practically haunts the title character — it's even played by a chamber orchestra at a posh party he is attending.
In Brazil, when Sam is entering a code into an elevator, eight notes of the theme song are played.
In one of the film's more affecting sequences, Michael Corleone reminisces about his time in Sicily as his son sings an Italian version of "Speak Softly Love ("Love Theme" from The Godfather)" in The Godfather Part III, the main theme is also heard being played by music bands in several scenes during the movies.
In Ghostbusters II, Ray and Winston play a tape of the "Ghostbusters" theme song at a kid's birthday party (the ungrateful little yuppie larvae scream for He-Man). The movie also has characters saying the song's line, "Who you gonna call?"
In the film Serenity, Jayne is strumming the theme to Firefly on his guitar when the crew first arrives on Haven.
In Tourist Trap, one of the secondary cues is played in "music-box" form when a switch is hit, lighting up a display of a wax mannequins.
Thor has a scene where "Walk" by the Foo Fighters, the movie's ending theme, is playing on a bar jukebox.
The main orchestral theme of The Ring appears in the actual film as a tune that Samara used to sing as a child. Unsurprisingly, it's creepy.
At one point in Dario Argento's Tenebre, a character puts on a record that turns out to be the film's theme music. Another character shouts at her to turn it down, but she ignores the request, and it fades up into the foreground to accompany the following scene.
The 2008 Get Smart film riffed on the famous opening sequence with Max going through a series of high-security doors with him not clearing one door in time. The theme music stops and he has to input a code, which happens to be the next couple bars of the theme.
In the first film, when Marty and his band are auditioning to play at the school dance, everybody but Marty plays the opening riff of "The Power of Love" by Huey Lewis and the News — the theme song for the movie.
In the remake of Land of the Lost, the theme is repeatedly referenced by lines like "This is the greatest earthquake ever known!" until Rick Marshall sings the show's entire theme song as something he made up. Never mind that he's referring to himself by his last name.
Also in Star Wars, it's implied in the Expanded Universe that the Imperial March is the Empire's martial theme. One officer is seen to whistle, "dum, dum, dum, dum-te-tum, dum-te-tum" in a Boba Fett comic.
In Rocky III, during the ceremony celebrating Rocky's statue the band is playing "Gonna Fly Now". It happens again at the beginning of Rocky V.
At the end of the first Tales from the Crypt movie, Demon Knight, a man implied to be a demon is shown walking down a desert road whistling the series (and the movie's) main theme.
In Labyrinth, the Nostalgic Music Box that plays when Sarah picks up the ballroom dancer figurine is "As The World Falls Down", which is played in its entirety later on. Not quite the theme music, but definitely a cameo.
The Saint theme, which goes back to the 1939 film The Saint Strikes Back and featured in the 1960's TV show, appears as a car alarm in the 1997 film during an early scene.
Gamera plays his own theme tune on Zigra's back after defeating him in 1971's Gamera Vs. Zigra.
In Help!, Clang and his thugs attempt to ambush The Beatles in an Indian restaurant, first nabbing the house musicians and replacing them, playing an instrumental medley of tunes from A Hard Day's Night (their previous movie) on Eastern instruments.
In the live-action Thunderbirds movie, Lady Penelope and Jeff Tracy's ringtones are the "Thunderbirds March".
In the Inspector Clouseau film A Shot in the Dark, the band at Camp Sunshine are playing the film's theme tune.
The Witches of Eastwick (1988) has a very catchy and distinct main theme, composed by John Williams. At one point Jack Nicholson's character is whistling the melody, as if he's able to hear the film's score (this is near the end, when he goes shopping).
In the pseudo-Christmas movie Gremlins, as the eponymous creatures carol at the door of the Scrooge-ish Mrs. Deagle, they are singing (read as: screeching off-key) the movie's theme song by composer Jerry Goldsmith.
A kind of meta example in the case of Galaxy Quest, in which the in-universe theme song of the eponymous show is a constant presence within the movie's own soundtrack.
The film features three in-universe renditions of the theme. The original 1970's show version, an electro-rock cover version played at the convention (which humorously announces Jason Nesmith's over the top entrance), and a reworked theme for the new TV show (basically the same as the original, but played by a much bigger orchestra).
Laura has the theme music playing on a phonograph in the title character's house.
In Agent Cody Banks 2: Destination London, Cody (undercover as a music student) is racing to return to his room, while his fellow students are practising their instruments in their rooms - and they happen to be playing Cody's (and the movie's) theme music.
In Mash, the "Suicide Is Painless" theme is performed by one of the attendees during the "funeral" for Painless Pole. As it and The Long Goodbye, above, were both directed by Robert Altman, it seems to be a director trademark.
In Veronica Mars (The Movie follow-up to the show), Veronica and Piz pass a street busker who is playing the TV show's theme song, which continues to play under Veronica's voice-over as she talks about the case.
In the second The Librarian film, a set of self-playing pan pipes started playing the film's theme while Judson was scolding Flynn.
In The Longest Day the pianist in the RAF mess is playing a slow version of the film's theme song.
In Spaghetti WesternA Stranger In Town (1967) the female antagonist is humming the theme tune while changing in her room; this happens mere four minutes after the same tune had been played during the opening credits. Strangely enough, it is never played diegetically again.
An odd example in A Fistful of Dynamite. Ennio Morricone composed a musical theme for each of the two protagonists, which is played in the background when we see them, but never diegetically. Then, all of a sudden, one of the protagonists begins to whistle "his" musical theme; but what really takes the cake is the fact the other one, not even seeing his buddy, instantly recognizes him by what's being whistled. A strange case of either Magical Realism or Medium Awareness.
Barbarella: The titular character hums the movie's theme song (obviously also hers) after having sex with Mark Hand and Pygar.
Twisted Nerve features the main character repeatedly whistling the title theme, composed by Bernard Herrmann. In fact, the whole soundtrack is the same tune performed in different fashions. Quentin Tarantino borrowed the tune for Kill Bill Vol. 1, where an assassin disguised as a nurse... that's right, whistled it.