I've been thinking of a particular kind of pun, which I can best describe using the quintessential example:

Have you heard? The Pillsbury Doughboy died. [Oh, how sad! How?] Yeast infection.

For want of a better term, I therefore call them "doughboys." The other example I know of is:

Have you heard? Cap'n Crunch died. [Oh, how sad! How?] Cereal killer.

No, I'm not claiming this is a high art form -- but I thought I'd open up the comments for people to add others. Remember, the goal isn't to mock someone (no matter how much they deserve it); it's just a pun, preferably a slightly absurdist one.

Paul Zrimsek (mail):
Have you heard? A thousand people just got killed in St. Louis! [My God! Terrorism?] Fallen arches.
8.3.2005 3:12pm
Two Stinkers:
Have you heard Arthur Fonzarelli died? Foot and Malph disease.

Have you heard Rhett Butler died? Scarlet fever.
8.3.2005 3:13pm
what the heck:
Have you heard? David Hyde Pierce died. [OHS! H?] West Niles.
8.3.2005 3:19pm
Hei Lun Chan (mail) (www):
Have you heard? Lou Gehrig died. [Oh, how sad! How?] Lou Gehrig's Disease.
8.3.2005 3:53pm
Bub (mail):
How are these different from the type of pun usually known as a Tom Swifty? I lost my toothpaste Tom said Aimlessly.
8.3.2005 4:04pm
Have you heard Jimmy Carter died? Oh, how sad. How?
Peanut allergy.
Have you heard Teddy Roosevelt died? Oh, how sad. How?
Run over by a bullmoose.
8.3.2005 4:08pm
charlieb (mail) (www):
The doublemint twins. Gum disease.

America's largest bagel tycoon. Noah's bleed.

Tony the Tiger. GrrrrrAVES! disease.

Wow. This is kind of fun.
8.3.2005 4:16pm
I believe the difference is that Tom Swifties are one-liners, whilst these are question and answer phrasings.

BTW, I think your TS would be sillier as:
Crestfallen, Col. Gate wandered aimlessly, searching for his toothpaste.
8.3.2005 4:19pm
Eh Nonymous (mail) (www):

Did you hear the Jolly Green Giant lost his job? ... Canned.

Have you heard the Volokh Co-Conspirator's case was lost after all? [Oh dear, how?] JNOV.

What's the sound a Volokh Conspirator makes when [a car hits them/ they get accidentally electrocuted/ someone cuts their throat/ etc.]? Take your pick: Kerr, Volokh, Korobkin, Zywicki, Post, Jaffe. Most of y'all look onomatopoeic anyway.
8.3.2005 4:24pm
Have you heard? The winetaster in the bodycast died. [Oh, how sad! How?] He got plastered in Paris.

Did you hear Marge "the Designated Hitler" died? [Oh, how sad! How?] She was Schott.

Did you hear the Scotsman died? [Oh, how sad! How?] He was kilt.

Did you hear Mr. Potato Head died? [Oh, how sad! How?] Weapons of Mash Destruction.
8.3.2005 4:35pm
uh clem (mail):
Did you hear Willie Nelson got run over by a car? (No, how'd it happen?) He was playing On the Road Again.
8.3.2005 4:41pm
Gary McGath (www):
Some of these wander pretty far from the mold or just aren't puns, so here's my attempt to enlarge the canon:

The Michelin Man died: Tired of living.
Dr. Frankenstein died: Too much eating hot dogs and drinking beer.
Frodo died: Ringworm.
8.3.2005 4:42pm
W J J Hoge (mail):
Doctor Who died--timed out.
8.3.2005 5:43pm
Noel Magee:
William of Ockham died: Slashed with his own razor.
8.3.2005 5:43pm
Devin McCullen (mail):
This doesn't fit, but it's close enough for me:

Did you hear Buckwheat became a Muslim and changed his name? (No, what did he change it to?) Kareem of Wheat.

I'm also reminded of a bit from my old Boys Life magazines with fake tombstones. The one that's always stuck in my head:

8.3.2005 5:45pm
James Ellis (mail):
My favorite one of these actually happened. I remember being amused by it as a kid:

Did you hear that Euell Gibbons died? Natural causes.
8.3.2005 5:50pm
Doug Sundseth (mail):
"Did you hear that Euell Gibbons died? Natural causes."

[A continuation] It's not a complete loss, though.

Why's that?

Many parts are edible.
8.3.2005 6:08pm
Did you hear Kermit the Frog died?

Oh no! How?

Animal attack.
8.3.2005 6:10pm
Tinhorn (mail):
On, reader-submitted headlines do this regularly when a person of some fame dies. Perhaps the most canonical was:

Al Hirschfeld dead at 99. Details sketchy.

One I submitted, which was greenlighted and made the main page, was:

Tony Roma dead at 78. Meat maker meets Maker.

I view these as signs of respect. Others' mileage may vary.
8.3.2005 6:19pm
Did you hear the inventor of the mechanical pencil died? Pumped full of lead.
Did you hear Ronald McDonald died? Big Macular degeneration problem.
Did you hear Martha Stewart died? Quit Living.
Did you hear David Koresh died? Unsafe sect.
Did you hear the 80s glam band died? Poisoned.
Did you hear Johannes Gutenberg died? Depression.
Did you hear E.O. Wilson died? Caught a nasty bug.
Did you hear Jane Goodall died? Naturalist causes.
Did you hear the urologist died? Just petered out.
Did you hear Oscar the Grouch died? Bodily wasting.
Did you hear Pluto died? Fell into your anus.
8.3.2005 6:43pm
Have you heard? John Holmes died. [Oh, how sad! How?] He was hung.
8.3.2005 6:46pm
John Beukema (mail):
One that I found uproariously funny at the age of about 10, many, many years ago:
Did you hear that Betty Crocker is in the hospital? She backed into a hot stove and burned her buns.
8.3.2005 6:55pm
Have you heard about the guy that fell in the lens grinder? Made a spectacle of himself.

Have you heard about my friends the human hinges? Couple of real swingers.
8.3.2005 7:42pm
Evelyn Blaine:
There are enough here to make you tired of this game forever:

Philosophers' Deaths

A few of the less recondite ones:

Adorno: Bad frankfurter
Anselm: Disease than which no deadlier can be conceived
Berkeley: Divine neglect
Camus: The plague
[etc., etc.]
8.3.2005 8:58pm
Evelyn Blaine:
There are enough here to make you tired of this game forever:

Philosophers' Deaths

A few of the less recondite ones:

Adorno: Bad frankfurter
Anselm: Disease than which no deadlier can be conceived
Berkeley: Divine neglect
Camus: The plague
[etc., etc.]
8.3.2005 8:58pm
Evelyn Blaine:
There are enough here to make you tired of this game forever:

Philosophers' Deaths

A few of the less recondite ones:

Adorno: Bad frankfurter
Anselm: Disease than which no deadlier can be conceived
Berkeley: Divine neglect
Camus: The plague
[etc., etc.]
8.3.2005 8:58pm
How do you destroy a vampire cow? [I don't know, but living in Nebraska, I sure would appreciate knowing] Sirloin stake.
8.3.2005 9:04pm
Barbara Skolaut (mail):
Did you hear about the cow who drank a bottle of ink?

Mood indigo.

This is my all-time favorite pun. :-D

Actually, it's only 2/3 of a pun.

(What's 2/3 of a pun? you ask)

8.3.2005 9:14pm
Stephen VanDyke (mail) (www):
Did you hear America can't find it's checks and balances?

No way, where could they be?

They could start looking in the Caymans.

ba dum tsshhhh
8.3.2005 9:30pm
C L (mail):
Did you hear that an ex-president died? Ray Gun.
8.3.2005 10:12pm
arthur (mail):
LAX just died? Terminal cancer.

And consider the celebrities whose obituaries write themselves, such as

Howard Keel-ed over.
8.3.2005 10:49pm
Michael Kleber (mail):
Have you heard? [Eats, Shoots &Leaves author] Lynn Truss just died.

Colon failure.
8.3.2005 11:06pm
Not PC:
Did you hear that Lemon and Lime just died? ...AIDS.
8.3.2005 11:08pm
triticale (mail) (www):
And then there was the cheesemaker's daughter who found herself in the family whey.
8.3.2005 11:44pm
Alan K. Henderson (mail) (www):
Did you hear that Casey Kasem died when a television fell from a high-rise window? Video killed the radio star.
8.4.2005 1:58am
CJ (mail):
Have you heard? Princess Diana died. [Oh, how sad! How?] Car accident.

Wait...I don't really get this thread.
8.4.2005 6:39am
Roy Lofquist (mail):
Did you hear what happened to Helene Curtis? Max Factor.
8.4.2005 10:20am
Archangel (mail):
After reading the above offerings, here's a comment and another offering:

Remember the terrible infection that almost destroyed the nation's strategic humor stockpiles when Knock-Knock jokes escaped containment - so please be careful with this new and virulent form.

and (also after reading these offerings):

How did the the tunnel driller die? Bored to death.
8.4.2005 10:29am
Gary McGath (www):
James Doohan is dead Jim.
8.4.2005 10:49am
Thief (mail) (www):
Did you hear about the lady who backed up into the fan?

8.4.2005 11:12am
NickM (mail) (www):
Albert Einstein died. Stepped in front of a speeding light.

Captain Hook died. Jock itch.
Captain Hook died. De-Capitation.

8.4.2005 12:40pm
David Rose (mail):
Hear about the butcher who backed up into the meat grinder?

He got a little behind in his orders.
8.4.2005 2:37pm
Mycin (mail):
A butcher backed into a meat grinder...he'll be OK, but he got a little behind in his work.
8.4.2005 2:39pm
Mycin (mail):
David just beat me to the wire on that one...
8.4.2005 2:40pm
David Rose (mail):
Sorry, Mycin.

This grade school stuff is excellent!

Hear that Betty Crocker was committed to a mental hospital?
She went stir crazy.

What did Della wear?
She wore her New Jersey.
8.4.2005 5:12pm
David E Bernstein (mail):
What did Della wear?
I don't know Alasker
8.4.2005 10:50pm
Raphael Laufer (mail):
Did you hear how the Third Man died?
Lyme disease.
Well, he did was suffering from Cotton-mouth.
Alls well that ends Wells
8.5.2005 12:25pm
Kev (mail) (www):
"What did Della wear?
I don't know Alasker

Shouldn't that be, "Idaho. Alaska." ??
8.7.2005 2:33am
Jackal (mail):
Well, I've always wondered: Juneau what the capital of Alaska is?

(We're kind of wandering off-topic here...)
8.7.2005 1:01pm