Dead Ringers managed to avoid this trope a surprising amount of the time despite being completely based on impressions and parodies, as most of its regular targets were also BBC productions. They did use Suspiciously Similar theme music for some of their one-off sketches, though, such as when they put one of the BBC's most well-known political interviewers into a superhero story to form The Continuing Adventures of Paxman.
American Top 40: A Suspiciously Similar version of "Afternoon Delight" (by the Starland Vocal Band) was used as a cue from 1977-1978.
Mitch Benn's songs on The Now Show are generally to tunes that sound like the songs he's parodying, since the UK doesn't have a "fair use" exception for parodies.
For a while during commercial breaks on Radio Disney the hosts talked over a song suspiciously similar to Rage Against the Machine's "Killing In the Name" (of all songs!).
Many radio station jingle packages will contain cuts that are very closely based on actual songs. For example, cut #20 from JAM Creative Productions' "Music Jam" package is an obvious tribute to The Jacksons' "Shake Your Body (Down To The Ground)". The similarity was lampshaded when this jingle was resung for New York's "Jammin' Oldies" station in the late 90's, as their version used a Michael Jackson soundalike vocalist.
Although not a musical example, Denis Leary's entire act was a Suspiciously Similar Version of Bill Hicks'.
In the musical Dames At Sea, "That Mister Man Of Mine" has a melody mostly copied from "The Man I Love".
The "Nightingale Lullaby" from the musical Once Upon a Mattress includes an obvious pastiche of the Lullaby from Stravinsky's ballet music for The Firebird. It's even written in the same key (and since that key is E flat minor, this is actually significant).
Used to dark comic effect in "My Psychopharmacologist and I" from Next To Normal, where a litany of antidepressant medications (and their side effects) is sung to the Suspiciously Similar Version of "My Favorite Things" from The Sound of Music.
The final refrain of "Letting Go"("not letting go, not letting go, never letting go, never letting go... of you") from Vanities: The Musical resembles the ending of "Friend Like Me" from Aladdin. The verse and pre-chorus of Hey There, Beautiful are a JHV of Mr. Blue Sky by Electric Light Orchestra, although the latter has a longer verse melody.
When a Web Animation series reaches a certain level of success (typically when it starts selling DVDs), the creator often goes back and removes any copyright infringement that was safe when the series was unknown. Bonus Stage is a good example: Matt eventually removed a multitude of unauthorized cameos from his earlier episodes (such as one by the Homestar Runner cast) and replaced the ska song in the credits with an instrumental facsimile called "Total Soundalike."
Speaking of Homestar Runner, one of the earliest Homestar toons ("Marshmallow's Last Stand") featured a snipet of the theme fom The Good, The Bad, And The Ugly. When the toon was released on DVD, the audio was changed to a JHV of the tune. Lampshaded by Matt Chapman (as Homestar) in the commentary when he tells Mike Chapman "don't listen to this part, we're gonna change it." when the original audio can be head in the backound.
Lampshaded somewhat in the cartoon "On Break", in which Mike and Matt perform an a-capella rendition of "Yakety Sax" that ends somewhat differently from the original song. Afterwards, one of them asks, "Does it end different?"
Another would be Zero Punctuation, which originally featured segments of various "appropriate" songs during the opening and closing credits. Lampshaded at the start of the first episode with the new theme music.
Because YouTube is so ridiculously litigious about the use of music in parodies, parodists like Venetian Princess now have to find someone like Steve Goldstein to write a Suspiciously Similar Version for them. Example here.
Used regularly on How It Should Have Ended to imitate the themes of the movies they're parodying, featuring "The Stuff" ("The Touch"), and the Terminator beat with an extra note.
The Guild payed homage to Reservoir Dogs in Season 3, Episode 10 "The Return". The Knights of Good carry their computers, slow-motion style, from Vort's van to the LAN battle, all to a Suspiciously Similar Version of "Little Green Bag". It is awesome and hilarious at the same time.
Ultra Fast Pony frequently uses copyright-infringing song snippets, but the one time the series does use a substitute (a cheap, a cappella substitute, at that) the characters themselves comment on it.
Twilight: What, we couldn't get the real music? Pinkie Pie: Oh, no way, bro! It's copyrighted! Twilight: Never stopped us before. Pinkie Pie: Oh, quiet, bro, or they'll hear you!