My Company asked Airbus that very question a few years ago and the answer was very specific. The "exceptional conditions" refer to conditions where "LAND ASAP" is displayed in red on the ECAM or when there are unambiguous indications that it is unsafe to continue the flight. In all other conditions the aircraft weight shall be reduced to maximum landing weight or less prior to landing.
With regard the A320 and A340, yes there is a graduated check to be done depending on the landing weight. The first step is straight forward and requires about 2 hours in the hangar. So even a minor overweight landing will have an engineering impact on schedule. In the case of the 340, is will be cheaper and quicker to dump the fuel before landing. From experience, landing a 320 at almost max take off weight put the aircraft in the hangar for more than 24 hours.
Have a nice day