A teddy bear riddle was posted on the newsgroup alt.tasteless.jokes on October 11, 1997:

Q: Why didn’t the teddy bear eat anything?
A: He was already stuffed.

A dessert version was printed in the Los Angeles (CA) Times on February 3, 2002:

Q: Why didn’t the teddy bear eat dessert?
A” Because he was stuffed.

The joke has been printed on several images.

“Who doesn’t eat on Thanksgiving?"/"A turkey because it is always stuffed” and “What did the turkey say before it was roasted?"/"Boy! I’m stuffed!” are similar “stuffed” riddles.


Google Groups: alt.tasteless.jokes
Stupidest joke ever. 
Mouse
10/11/97
no no this one is even dumber....
why didn’t the teddy bear eat anything?
he was already stuffed

Google Groups: alt.tasteless.jokes
Any jokes about big ears?
Steve
8/24/98
(...)
Why didn’t the Teddy Bear eat anything?
Because he was stuffed.

3 February 2002, Los Angeles (CA) Times, “The Kids’ Reading Room; Jokes & Riddles,” pg. E6, col. 4:
Los Angeles Times; Los Angeles, Calif. [Los Angeles, Calif]03 Feb 2002: E.6.
Why didn’t the teddy bear eat dessert? Because he was stuffed. (David, 6, Hancock Park Elementary, Los Angeles)

Google Groups: alt.humor.puns
Puns of the Weak: 02/08/02
Stan Kegel
2/8/02
(...)
Why didn’t the teddy bear eat dessert?
Because he was stuffed. (David, 6)

Google Books
Lunchbox Laughs:
A Book of Food Jokes
By Mark Ziegler
Minneapolis, MN: Picture Window Books
2005
Pg. 5:
Why didn’t the teddy bear eat dessert?
It was stuffed.

Twitter
OurLady‏
@OurLady
Why didn’t the teddy bear eat dessert? . . . . . Because he was stuffed.
6:16 AM - 6 Oct 2008

12 October 2008, Sunday Territorian, (Darwin, Australia), “Jokes,” pg. 32:
WHY DIDN’T THE TEDDY BEAR EAT HIS DINNER? Because he was stuffed!

Graphic Joke
Teddy bear dessert…
May 31, 2012 by grin
Q: Why didn’t the teddy bear eat dessert?
A: Because he was stuffed!

Twitter
Ellen DeGeneres‏
@TheEllenShow
Why didn’t the teddy bear ask for seconds? He was stuffed #ClassicJokeTuesday
4:40 PM - 13 Nov 2012

20 November 2012, Birmingham (UK) Mail, “Chipper Jokes,” pg. 33:
Why didn’t the teddy bear eat all of his dinner? Because he was stuffed!

Twitter
Laffy Taffy Jokes‏
@Laffy_TaffyJoke
Why didn’t the teddy bear finish his supper?
Because he was already stuffed.
8:24 PM - 5 Mar 2014