In order to make this work, they need to insulate the tubing so the beverage remains hot on its journey.
From the Letterman site, via Starbucks Gossip, which, curiously is run by Jim Romenesko of the vastly more famous media site:
Dave is thirsting for some Starbucks coffee and tonight, to satisfy his desire, we rigged up something really special. From the Starbucks across the street and down the block, we have a direct link via 550 feet of clear plastic tubing. Dave has a spigot at his desk. The source is in Starbucks. The power to get the coffee from Starbucks to Dave's desk is supplied by a nitrogen tank at Starbucks. We turn on the camera at Starbucks and meet and greet Brad Simanski at the counter. ...
When all is ready, Brad the Barista turns on the power and Dave's decaf coffee is sent on its way. The camera follows the coffee leaving the n/e c/o 54th and Broadway. Across 54th is goes, then across Broadway, through the Ed Sullivan Theater doors, through the lobby, down the side of the theater and to the spigot. Dave turns on the faucet to enjoy a nice delicious cup of Starbucks. Complains the customer; "It's too cold." This technology is still in its infancy stage and portions still need to be worked out. Over all, though, a success. The coffee from Starbucks was a success. Big money was lost on this bit. No, not on the creating of the whole thing . . . but on the money bet that it wouldn't work. It was rehearsed once with a modicum of success. For the show, we were very happy with the results.