Definitely. Check out The Bobs, for instance, or DaVinci’s Notebook. Lots of college groups do some humorous things as well. The Best of College Acapella has a humor album out called Wasting Our Parents’ Money: The Best of College Acapella Humor. Great stuff.
Slightly more arguably-novelty…
Oh my God, Becky… did you see her butt?
Be grateful you escaped the Fast Food Rockers 
There are at least two popular extant bands besides Weird Al still making novelty songs – Ween and They Might Be Giants. True, TMBG wants to extend their recognition beyond the “joke/humor” song, but if either of these bands were released when “novelty” was considered a viable genre, they’d be considered novelty.
That would be by the truly bizarre Nervous Norvus. We used to have an original 45 of that song on the Dot label…unfortunately gone with the years.
Well, who knew? Didn’t know about the Zappa connection - I thought the song was about Jerry Lewis. Shows what I know.
The Violent Femmes have some songs that I’d consider novelty. They’re good - check 'em out.
A Pizza Hut, A Pizza hut
Kentucky fried chicken and a pizza hut.
ARRRRGHH!!!
Another slight hijack, but what became of Julie Brown? I know she had at least two novely pseudo-hits with ‘Because I’m a Blond’ and ‘Homecoming Queen’s Got a Gun’.
25 All-Time Novelty Hits (2002 Varese Saraband) on shelf at Circuit City today:
The Flying Saucer (Pts. 1 & 2) -Buchanan & Goodman
Western Movies-The Olympics
Baby Talk-Jan & Dean
Nee Nee Na Na Na Na Nu Nu-Dicky Doo & the Don’ts
Alley-Oop-Hollywood Argyles
Jolly Green Giant-The Kingsmwn
Juanita Banana-The Peels
They’re Coming to Take Me Away Ha-Haa-Napoleon XIV
& 17 more. Really great.
Buchanan & Goodman dubbed phrases from other pop songs into their Flying Saucer opus narration. Got sued for copyright infringement my several owners, I remember reading.
I suspect that the latter tune just doesn’t play as well today, after all of those real-life school shootings, as it did when it first came out 
And Yankovic is a genius, pure and simple. One important key to his lyrics is the subtlety. I can hear one of his songs 50 times, and on the 51st listening I’ll catch some little thing I had missed before, and it will start me howling all over again.
For example:
In Fat, he replaces Michael Jackson’s “Jam on! Jam on!” with “Ham on! Ham on! Ham on whole wheat!” That becomes insanely funny once you realize that the Spanish word for “ham” is “jamon”. Pronouned “ham-on”.
The number “27” is included in the lyrics at least once on every single album Al has ever released. Not exceptional at first glance. Even Al himself has stated that he just thinks 27 is a funny number. But then one day you’re looking at the Periodic Table, and you discover that 27 (actually 26.981538) is the atomic weight of the element Aluminum, which is indicated by the symbol “Al”.
I can’t tell you how many times I played, yes, even performed in public, Al’s song One More Minute, before I finally realized what I was saying when I sang the line, “'Cuz I’m stranded all alone at the Gas Station of Love, and I have to use the self-service pump…” I finally, painfully, agonizingly realized the meaning immediately after I sang that line… to a group of people from my church. :smack:
It’s especially funny when, while singing about some horrifying disaster, he expresses concern over the most trivial thing. Singing about the Earth being invaded by space aliens who are bent on killing everybody, he sings, “They’ve got hands all covered with fungus / They’ve got eyes like some kind of bug / I sure hope they don’t come in here / I just shampooed the rug!”
*I’m the biggest dork there is alive
My mom picked out my clothes for me 'till I was 35
And I forgot to mention
I’m not even welcome at the Star Trek convention
But the Frenchies think
That my poop don’t stink
I’m a genius in France
*
Fish heads, fish heads,
roly-poly fish heads,
fish heads, fish heads,
eat them up, yum!
How about Bob Rivers?
Both Louden Wainright III (“Dead Skunk”) and Jim Stafford(“Cow Patti”, “My Girl Bill”, “Cows with Guns”) had big hits with novelty songs in the 1970’s. John Prine sang “Dear Abby”, and Zevon had “Werewolves of London”. So there have been a few around. But I definitely think Weird Al qualifies.
Didn’t Buchanan & Goodman do about a half-dozen of those things (I vaguely remember one about the shark in JAWS)? Then they fell off the map, and now we know…
the rest of the story.
When I was a kid, I loved My Boomerang Won’t Come Back by Charlie Drake.
And, I actually preferred The Martian Hop to the Beach Boys. :o