Such, at least, is the claim of the World Wildlife Fund video here, and of the post at Green Planet linked above. We wouldn't presume to evaluate the science behind it.
The obvious suggestion, which both the WF and GP make, is to skip the disposable cup, napkin and cuff -- ask the barista to fill the mug you brought from home.
We have another suggestion, however, which seems even more obvious, at least to us: Don't try to drink your coffee while you walk or (ye gods!) drive. It's absurdly unsafe, for one thing, and one more concession to the spirit-killing age of speed and portability for anther.
If you're having coffee away from the collection of mugs you keep at home or at work, don't plan on takeout. Sit down, or stand at the bar like an Italian, and take a couple of minutes to savor the rush of caffeine. Then put down your ceramic cup and walk away, your head buzzing. And if you're running late but still desperate fro an eye-opener, then you can really copy the Italians, and pound a double espresso. Six seconds and off you go, a happier person in a greener world.