“Teepee” (a cone-shaped tent) sounds like “tea pee.” A tea riddle is:
   
Q: What happened to the Indian chief who drank too much tea?
A: He drowned in his tea-pee.
 
The joke was posted to the newsgroup alt.usage.english on October 13, 1998.
 
         
Wikipedia: Tipi
A tipi (also tepee or teepee) is a cone-shaped tent, traditionally made of animal skins upon wooden poles. A tipi is distinguished from other conical tents by the smoke flaps at the top of the structure. Historically, the tipi was used by Indigenous people of the Plains in the Great Plains and Canadian Prairies of North America, as well as by indigenous peoples of northern Europe and Asia under other names. Tipi lodges are still in use by these peoples, though now primarily for ceremonial purposes.
 
Google Groups: alt.usage.english
Question on jokes regarding Indians.
Lars Eighner
10/13/98
(...)
3. “Then there was the Indian who preferred tea to
|      firewater. Unfortunately, he drank too much one night and drowned
|      in his tea pee.” How is this one supposed to be funny?
 
Beats me why anyone would think this is funny.  It is supposed to be a play on words:  “tee-pee” = lodge, “pee” = urine. Ha. Ha. Ha.
 
Google Groups: alt.underground
leaping fry daddy
Cybernetic Tiger
6/15/01
(...)
Tipi = little tent that some native Americans live in
Tea-pee = urine produced from drinking too much tea
 
Google Groups: alt.humor.puns
Puns of the Weak 6/29/01
Stan Kegel
6/30/01
(...)
Did you hear about the Indian who drank 12 gallons of Lipton’s? They found him the next day, dead in his tea-pee. (Keith Suchland)
   
Google Groups: rec.humor
T Day
nemo
6/25/03
(...)
Tea-Pee: Having to leave one’s tent to relieve oneself after drinking too much tea!
       
Google Groups: news.admin.net-abuse.email
A good Web host who won’t spam
Norman L. DeForest
3/12/04
(...)
> > /me hands Morley a hot cuppa ‘T’.
>
> Thanks.  Can’t have too much “t” can ya?
 
Well, there was the case of the Indian who drank too much of it.
They found him one morning, drowned in his tea pee.
 
Google Groups: alt.humor
Re: Longest Thread Ever    
nemo
1/16/05
(...)
> Tea Pee: What happens if you drink too much tea.
>
The smell was intense?
 
Google Groups: rec.humor
Definitons  
Tim Bruening
3/7/05
(...)
Tea Pee: What happens if you drink too much tea.
 
Google Groups: alt.humor
Tea Puns
nemo
6/21/05
(...)
Tea-Pee: What you do in a tent when you’ve drunk too much of the stuff!
 
Twitter
Jen Castle Photo‏
@JenCastle
I’m starting the day off with a joke- what happened to the little Indian that drank too much tea? . . . He drowned in his Tea-Pee !
11:39 AM - 4 Nov 2009
 
Twitter
Darren Gooding‏
@dazzybabes
Replying to @mariecurieuk
@mariecurieuk What happened to the Native American who drank too much tea? He drowned in his Tea-pee 😊
5:10 AM - 15 Jul 2010
     
Reddit—Jokes
What happened to the Indian chief who drank too much tea?
submitted November 29, 2013 by michaelscarnfbi
He drowned in his tea-pee.
 
Reddit—Jokes
An oldie but goodie. What happened to the Indian who drank too much tea?
submitted February 24, 2014 by [deleted]
He drowned in his tea-pee. EDIT: *Native American, as less oblivious people have pointed out.
 
Twitter
thankful dad jokes‏
@dadtellsjokes
What happened to the Indian who drank to much tea? 
He drowned in his tea-pee
8:15 AM - 3 Oct 2017