Both dentists/orthodontists and lawyers have “retainers.”
     
“A lawyer asked his dentist to give him a retainer. (Pun of the Day)” was posted on the newsgroup rec.humor.jewish on June 24, 2001. The joke originated on the website PunoftheDay.com.
   
“What do the teeth pay their lawyer? Retainer fee!” was posted on Twitter by Dentist Appointment on May 24, 2016. “How does an orthodontist hire a lawyer? He puts him on a retainer” was posted on Twitter by Aaron Buitron on July 13, 2019. “How do dentists pay for their lawyers? Retainers. #LaughoftheDay” was posted on Twitter by Jimmy RevJim Olsen on October 14, 2021.
   
         
Wiktionary: retainer 
Noun
retainer (plural retainers)
1. Any thing or person that retains.
2. A dependent or follower of someone of rank.
3. A paid servant, especially one who has been employed for many years.
4. A fee one pays to reserve the other’s time for services.
5. (dentistry) A device that holds teeth in position after orthodontic treatment.
 
Google Groups: rec.humor.jewish
Puns of the Weak 6/22/01
Stan Kegel
Jun 24, 2001, 12:59:01 AM
(...)
A lawyer asked his dentist to give him a retainer. (Pun of the Day)
     
Google Groups: alt.humor.puns
Puns of the Weak 06-14-03
Stan Kegel
Jun 13, 2003, 9:08:38 PM
(...)
What did the lawyer get at the dentists office?
A retainer (Stan Kegel)
 
Twitter   
Jim Chaparas
@JimGC
A lawyer asked his dentist to give him ..... a Retainer….....
12:21 PM · Jul 22, 2009·Twitter Web Client
   
Twitter

Diane Askwyth
@askwyth
A lawyer asked his dentist to give him a retainer.:+)
5:07 PM · Aug 23, 2009·Twitter Web Client
         
Twitter
Pete C
@peteec
“I have a lawyer on a retainer” - a sketchy orthodontist
7:10 PM · Apr 7, 2011·Twitter Web Client
 
Twitter
Mark T.
@Beercraft
My lawyer used to be an orthodontist. I have to pay him a retainer.
1:32 PM · Oct 21, 2011·TweetDeck
 
Twitter
Rosa Lei Steinard.
@Lensman23
Pun of the Day:  A lawyer asked his dentist to give him a retainer. #funny #jokes #laughs #humor
10:43 AM · Dec 5, 2012·Twitter Web Client
   
Twitter
Dentist Appointment
@Time4DentistPT
What do the teeth pay their lawyer? Retainer fee! #toothjokes PS It’s time to go to the dentist!
5:30 AM · May 24, 2016·Time4DentistBot
   
Twitter
Aaron Buitron
@apbuitron
How does an orthodontist hire a lawyer?
He puts him on a retainer.
10:55 PM · Jul 13, 2019·Twitter Web App
 
Twitter
Brew_SSsss
@Brew_SSsss
Replying to @PaulDoroshenko
Does the tooth fairy know that the dentist is like a lawyer? Both require a good sum of money for retainer fees.
1:41 AM · Sep 9, 2019·Twitter Web App
 
Twitter
Dentist Appointment
@Time4DentistCT
What do the teeth pay their lawyer? Retainer fee! #toothjokes PS It’s time to go to the dentist!
3:30 PM · Sep 29, 2021·Time4DentistBotCT
 
Twitter
Jimmy RevJim Olsen
@mybuddyjimmy
How do dentists pay for their lawyers? Retainers. #LaughoftheDay
7:07 PM · Oct 14, 2021·Twitter for iPhone