The glass jar should be mostly residue-free due to the smooth surface.
Sometimes slight traces remain when the jar is hand-washed but the harsher detergents usually used in dishwashers should get rid of them easily. (Side note: Don't do this to the coveted stained teapot of a true tea aficionado...)
The other parts are a different issue: often, they are not dishwasher safe and there are plenty of tiny nooks and crannies where coffee residue might "hide" and cling to and impart the characteristic coffee flavour - especially in combination with alcohol, which acts as a solvent.
If you are planning on using only the jar for your martinis, run it trough the dishwasher, give it an extra rinse with clear water just to be sure there is no trace of detergent and cheers!