"Istanbul (Not Constantinople)" was originally a Four Lads song, but most people these days only know the TMBG version.
"Dog on Fire", the instrumental theme song for The Daily Show was originally written and performed by Bob Mould. However, once Jon Stewart became the host a few years into the show's existence, Mould's version was replaced by one performed by They Might Be Giants, which has remained ever since.
"New York City" is a cover of a song by the all-female twee band Cub, whom the Johns were friends with. The original appears on Cub's 1994 album Come Out, Come Out, which had only been released two years prior to TMBG's cover of the song, which helped the fact that more people remembered the cover rather than the original.
"Why Does the Sun Shine? (The Sun Is a Mass of Incandescent Gas)" is a cover of a 1959 song by Tom Glazer. The song was woefully obscure when TMBG covered it and actually features a handful of now-inaccurate facts (notably that the sun is actually made of plasma, not gas). TMBG recorded "Why Does The Sun Really Shine?" a factually accurate new version of the song for their 2009 childrens' album Here Comes Science.
Epic Riff: "The Lady and the Tiger", in full effect.
Ear Worm: Many of these in their songs. Such as "Istanbul Not Constantinople".
Face of the Band: John Linnell and John Flansburgh (the original two members) are the ones that all fans of the band think of, although they have other members backing them up, many of them named Dan.
Nightmare Fuel: Hide Away Folk Family, especially the coda, which culminates in something not unlike Courage the Cowardly Dog suffering from catatonic shock.
Oh, you know that song "The Statue Got Me High"? The one with the statue that just makes people burst into flames and die? The song mentions it's being displayed publicly.
To add to that, try not to scare yourself with John Linnell's intense stare. Who says statues can't move. And sing.
"Cloissone". The narrator of the song's obviously up to some unsettling activity. "You've got a friend in law enforcement, don't go calling law enforcement..."
"Hide Away Folk Family", "Where Your Eyes Don't Go", "Someone Keeps Moving My Chair", "It's Not My Birthday", "Hall Of Heads", "A Self Called Nowhere", "The Bells Are Ringing", "Rat Patrol", "Older", "Ant", "Bastard Wants To Hit Me", "I'm Impressed"...
In case you're wondering, that's a song off of each (non-children's) studio album (and at least one compilation). Yeah, it's one of the band's favorite tropes. And at least half of those have serious Lyrical Dissonance. "Someone Keeps Moving My Chair" is downright upbeat.
Sequel Displacement: By far, even to this day, the most well-known album the band has put out is its third, Flood. This is the album that has "Istanbul (Not Constantinople)", "Particle Man", and "Birdhouse In Your Soul". Since it's the most common gateway into fandom of the band, there's even a Fan Community Nickname for those that bought Flood as their first TMBG album: Floodies.
They Changed It, Now It Sucks: In their early days, the Johns worked by themselves and used double-tracking and drum machines to complete the tracks, and they'd play to recordings live. When they finally decided they needed a backup band, fans were pissed. They got over it eventually, though, and now it's hard for most of them to imagine what it would have been like if that had never happened. Their first band album, John Henry, has a controversial status among fans.
What Do You Mean It's Not for Kids?: Even with some rather esoteric lyrics, it's pretty obvious that many of their songs touch on more mature themes. Yet their songs have appeared in multiple kid-friendly places (such as Tiny Toon Adventures and the first Power Rangers movie) even before they specifically made children's albums.