"The law of physics says you won't get something for nothing. In this case you might get improved economy but you might also find that your car doesn't drive as it once did. So for example you might find a flat spot in the power band or you may lose bottom-end acceleration." Quote from AA
On a diesel, the most efficient combustion point is 'TDC' (top-dead-centre). Typically the ignition has to be delayed a bit to pass NOx regulations. So one of the effects of a proper* remap is to move ignition back to this point.
Plus points:
* Better fuel consumption
* More torque
* Less particulate emissions
Negative points:
* Higher NOx emissions
* More strain on the clutch/gearbox
But claiming 20/30/40% improvement in fuel consumption is crazy, it's going to be a few % at best. If you see anything advertised with that sort of wild claim, steer well clear.
* By 'proper' I mean a remap which is unique to that individual car, since every engine is slightly different in terms of wear etc. If you have a 'chip', it is not tuned to that vehicle, and may smoke etc.
Audi A2 1.4 TDI w/6-speed gearbox. Best tank 87mpg, average of all tanks 75mpg. Fuelly ID is MMUK.