"Caffeine can be removed from coffee by treating the green beans with chlorinated hydrocarbon solvents."
"Coffee," Microsoft(R) Encarta(R) 96 Encyclopedia. (c) 1993-1995 Microsoft Corporation. All rights reserved. (c) Funk & Wagnalls Corporation. All rights reserved.

Presumably this means caffeine (like many other biochemical substances) is soluble in common organic solvents (like benzene, carbon tetrachloride etc.)
and can be dissolved out.
Ethyl acetate is also used as a solvent.
Hopefully, whatever solvent is used, the residual is well washed (as these solvents can be quite unpleasant)!
The caffeine can also be removed/extracted without the aid of organic solvents, but purely using water - as in a classic "steam distillation".
An example is "Nochem" (patent) method.
Water washing has the disadvantage, however (as well as added expense) of being a more destructive process (because of higher temperatures/longer heating time etc.), so it is more difficult to keep the subtler flavours of the tea or coffee.