Flaming Lips have some bizarro song titles and topics as well.
Madness have sung songs about:
Apartheid
School days
A man having a heart attack
Buying your first condom
An old, clapped out, but much loved car
A single mother with a mixed race baby
A happy family
an I.R.A. informant
Life in a borstal
Excuses offered by a burglar caught in the act.
Homeless people.
It was Paul McCartney’s aunt who asked him, “How come none of your songs are ever about anything?” and inspired him to compose “Paperback Writer.”
Phil Harris and his band, popular in the '40s and '50s almost never sang songs about love. They sang about smoking, playing cards, fishing, the South, the Thing, etc.
Maybe not their entire bodies of work, but some singles that fit the catagory:
Soul Coughing
Unmarked Helicopters
The Coffee Song
Nick Cave and the Bad Seeds
Red Right Hand
Brick
Dazz (disco-jazz)
Sparks is as quirky as it gets, but c. 95 % of their massive catalogue deals with women, sex and love, although often in disguise.
Nitpick: This one’s a cover, with many versions before TMBG, though I agree that Giants turned in the definitive version.
And to offer something new, and not just be a drive-by dick:
Mojo Nixon has a lengthy and varied oeuvre, including songs about barbecue, Elvis, Nuclear fallout, getting drunk alone, getting drunk with Jesus, getting drunk then committing stupid crimes, marijuana (and the legalization thereof), and one of my personal favorites, “Debbie Gibson is Pregnant with my Two-Headed Love Child,” though it is a love song of sorts. 
Big Audio Dynamite’s first record doesn’t seem to have any love songs, just lots of social commentary and movie references. “E=mc2” is about the movies of Nicholas Roeg.
I would add Rush to this list. Even their love songs aren’t really love songs.
Oops missed it was already there. I will second them then.
The Pixies: they may not be the best band to add to the list because of the general incoherence of their lyrics (typically, a song will have a key phrase or line that was its inspiration, surrounded by a lot of stream-of-consciousness or surreal word-salad imagery); but their discography tilts towards themes of environmentalism (and marine pollution esp.); man’s relationship to God (with some Hebrew numerology thrown in); Bible stories; and people acting in bizarre and (often sexually) aberrant ways.
Aside from those themes, though, they’ve also done songs that referenced the French engineer Alexandre Eiffel; matriculating at U. Mass; an alien searching for the planet that invented rock music; the films of the Spanish Surrealist filmmaker Luis Bunuel; and a song by Charles Manson (“Cease to Exist”) that was adapted by The Beach Boys.
And a quick-and-dirty writeup for post-Soft-Boys **Robyn Hitchcock **[if anyone feels like re-doing it better, please do!]: all sorts of offbeat topics, including trains, a widower who sees his wife’s ghost; failed parenting; churches, homoerotic body-building mags, and more trains. From the Wiki on RH: “Hitchcock’s musical and lyrical styles have been influenced by the likes of Bob Dylan, John Lennon and Syd Barrett. Hitchcock’s lyrics tend to include surrealism, comedic elements, characterisations of English eccentrics, and melancholy depictions of everyday life. His themes include the eternal contingencies of existence — namely: death, sex, and eating. (Recognising these themes, he released an EP in 2007 called Sex, Food, Death and Tarantulas.)”
Bloody your hands on a cactus-tree, wipe it on your dress, and send it to me… :o
Stereolab: Marxist social commentary, the anti-Rush, if you like.
Yeah, there are a number of big name acts who don’t have a lot of love/sex songs but aren’t really “quirky” either – they write about political and social issues, personal struggles with substance abuse or mental illness, etc. Aside from singer-songerwriters of protest songs, many or most rap, heavy metal, punk, hard rock, alt rock, and even country songs are not about love or sex.
If we distinguish romantic love from spiritual love then there’s also every song ever written about God or religion.
Kate Bush.
Such Tragically Hip songs as you can figure out the meaning to are quite often about, well, almost anything. Heroism (Fifty Mission Cap), Randomness and despair (Courage), Serial killers (Locked in the Trunk of a Car), and who knows what else. They do good love stories too, but it’s a mix.
I don’t think many serious musical acts do ALL their stuff about love and sex, do they?
Pere Ubu
- while you will find some about relationships (e.g. album titled “Why I Hate Women”
) it’s mostly about industrial USA (earlier stuff)
Alice Cooper. I mean, he sings about dead babies, going insane, being a cross-dressing killer, to name just a few off the top of my head.
Al Stewart. His biggest hit was “Year of the Cat,” which could be called a somewhat quirky love song (or maybe not); but a great deal of his output dealt with historical themes–see especially the Past, Present, and Future and Modern Times albums, although Year of the Cat itself includes a few historical songs as well.
Klaatu. Yes, there were some sappy love songs (“Dear Christine”), but much of the band’s output explored vastly different topics. Off the top of my head, I can think of UFOs (“Calling Occupants of Interplanetary Craft”), New York’s first subway (“Sub Rosa Subway”), and a boring day at work (“A Routine Day”). Some of their lesser-known songs simply defy categorization (“Silly Boys,” and “Little Neutrino”), and the one concept album they made (Hope) was about a dead planet.
I think a lot of early progressive rock would fit the OP’s parameters. Again, just off the top of my head, I can think of the Moody Blues’ Days of Future Past; which, although it includes “Nights in White Satin,” dealt mainly with the events of a single day. Genesis’ Foxtrot included some social commentary in “Get 'Em Out by Friday,” and a “personal journey” (as described by Peter Gabriel) in the 22-minute-and-change “Supper’s Ready.” There are other fitting works by these (and many other) prog-rock bands, which I won’t list, but if the OP is interested, he or she may wish to look into them.
Good point; I was really overreaching (and oversimplifying) in the OP.
And thanks, Only Mostly Dead, for the “Constantinople” correction – I had no idea!
Foxtrot also has “Time Table” which is about Western cultural self-importance or something.
Actually, pretty sure “Supper’s Ready” is the only love song on that album, & it has some kind of apocalypse in the middle.
Captain Beefheart. I win.
The Bonzo Dog Band. Song subjects include:
A statue (“The Equestrian Statue”)
A farm and its animals (“Jollity Farm”)
People going all out to look cool (“Look Out There’s a Monster Coming”)
An Elvis pastiche about a girl being murdered by a cab driver (“Death Cab for Cutie”)
An insane introduction to the members of the band – both actual and fictional (“The Intro and the Outro”)
Film noir (“Big Shot”)
Boredom (“I’m Bored”)
A dull seaside resort (“Postcard”)
Falling in love with an alien (“Beautiful Zelda”)
A parody of protest songs (“Can Blue Men Sing the Whites?”)
Humanoids (“Humanoid Boogie”)
A very annoying neighbor (“My Pink Half of the Drainpipe”)
Random and strange stories spoken over a musical background (“Rhinocratic Oaths”)
Shirts (“Shirt”)
An Ali-Baba parody (“Ali Baba’s Camel”)
Bodybuilding (“Mr. Apollo”)
The weirdness of touring (“The Bride Stripped Bare (by ‘The Bachelors’)”)
A Tent (“Tent”)
Constipation ('The Strain")
Bad personal hygiene (“King of Scurf”)
There’s also