It is called a "sleeve."
Although the primary meaning refers to that part of clothing which covers the wearer's arms in part or in whole, by analogy
it has come to mean a tightly fitting, tubular protective covering.
See Merriam-Webster for a more detailed definition.
ell.stackexchange.com/users/64237/Jeff Morrow
A historic term for a coffee cup sleeve would be zarf -- it traditionally refers to nondisposable Turkish metal sleeves but has also been used to refer to the disposable paper ones.
(Disclaimer: I've only once heard a sleeve actually being referred to as a zarf at a coffeeshop, and it was a "hey did you know" kind of thing from the barista)
ell.stackexchange.com/users/62062/Jimmy
It is called " Coffee cup sleeve ", and it is known also in the following names: coffee sleeves
Grip cup sleeves Coffee clutches Coffee cozies (refers mainly to those that they're for multiple use)
Hot cup jackets Paper zarfs Coffee collars
Hot cup holders ( Source ) Searching these names on internet stores prove that the most of them are in use in the market.
By clicking on the links you will see the results in Google images, and there you will find a lot stores which use these names.
ell.stackexchange.com/users/12430/Judicious Allure
I've certainly heard these corrugated cup-coverings casually called coffee condoms by college kids around campus (with the occasional “ribbed for her pleasure” snickers).
As might be expected of a coinage like this, Urban Dictionary offers the most baldfaced definition of this bit of street slang: Coffee Condom
The name for the sleeve that goes around your handle-less paper coffee cup to insulate the drinker's hands from hot coffee.
Similar to how a latex condom is designed to protect you from sexually transmitted diseases
the "coffee condom" is designed to protect your hand from the hot cup of coffee.
Java Jacket Smithsonian Magazine
from 2013-08-13 had an article on “How the Coffee Cup Sleeve Was Invented” containing a more detailed history of this device, including this small bit:
Jay Sorensen invented the Java Jacket in 1991 as a solution to a common problem—hot coffee burns fingers.
The idea emerged in 1989 when he was pulling out of a coffee shop drive-through on the way to his daughter’s school and a coffee spill burned his fingers, forcing him to release a scalding cup of coffee onto his lap.
At the time, he was struggling as a realtor in the years since closing his family-owned service station in Portland, Oregon.
While the coffee accident was unfortunate, it gave him the germ of an innovative idea: there had to be a better way to drink coffee on the go.
Sorensen ended up patenting his Java Jacket idea, and he sells a billion (B-I-L-L-I-O-N) of these a year.
But the one you get at Starbucks is ever so lightly different enough that they got their own patent after Sorenson got his.
An earlier patent for such a device was granted back in 1964:
And there have been many others besides that one.
The Smithsonian article also mentions one called a portable coaster back in 1947.
ell.stackexchange.com/users/141/tchrist
On the the more technically accurate side, the complete term would be coffee cup sleeve .
It's surprising that no one here actually mentioned that it even has got its own Wikipedia page.
Follow this link for the details.
ell.stackexchange.com/users/6450/Michael Rybkin
A “warmth protector thingy” can also be called an insulator , but sleeve , coffee sleeve , or even cardboard sleeve are all acceptable.
Coffee condom should be used very selectively.
ell.stackexchange.com/users/66350/B.W.
I recently got into an awkward situation when I wasn't able to ask for a 'warmth protector thingy' in Starbucks, and wasn't accurately able to describe it, so could someone tell me what you would call the brown paper thing covering the plastic cup of coffee? I struggle to find it anywhere because it's hard to describe, hence why the title is badly phrased. edit: would zarf be socially acceptable to use in a casual environment?
What is the brown cardboard covering coffee cups called in English?
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ell.stackexchange.com/questions/149863
ell.stackexchange.com/users/66440/Tyler
On the the more technically accurate side, the complete term would be  coffee cup sleeve . A “warmth protector thingy” can also be called an  insulator , but  sleeve ,  coffee sleeve , or even  cardboard sleeve  are all acceptable.    coffee sleeve , or even  cardboard sleeve  are all acceptable.   
A “warmth protector thingy” can also be called an  insulator , but  sleeve ,  coffee sleeve , or even  cardboard sleeve  are all acceptable.   