My thanks to former Mudcatter Azizi Powell,
who will not post here herself
but who kindly responded to my e-mail.

Azizi directed me to DARE,
the online Dictionary of American Regional English
at the website for
The University of Wisconsin-Madison.

An entry there reads:
tea for the fever, not to take
Not to put up with any nonsense.
Theresa is a woman with a "take no tea for the fever" attitude.

Azizi confesses in her response to me,
"I had never heard that saying
'not to take tea for the fever' before.
I asked my adult daughter and
she hadn't heard it before either.
[. . . ]
Perhaps it used to be commonly said
among some Black Americans way back when,
maybe in certain regions of the USA -- and maybe not."

and further:
"My GUESS is that
that saying came about because
herbal tea was one of the first options for a fever,
but when that didn't work,
people had to move to stronger methods
(perhaps such as hot toddies -- adding liquor to the tea)."