Wikipedia claims, without attribution, that a tea cosy:
... insulates a teapot, keeping the contents warm
The purveyor's of these teapot covers follow a similar vein:
Pretty, original and practical, any one of our tea cosies will keep the pot perfectly warm while lending a cheerful air to the tea table.
and even The Independent newspaper's got in on the action (albeit on what seems to be a piece written solely to push the things):
And if you're going to have a drop ("go on!"), you'll be wanting a nice tea cosy to keep it warm.
I find it hard to believe that a standard knitted/cloth tea cosy, with a bit of air to help out, will keep a pot of boiled water "warm". Do tea cosies make any perceptible difference to the amount of time a teapot remains warm?
The definition of warm is, of course, up for grabs.