Apart from the fact that hotter air has lower density and thus produces less lift on the airframe, I believe that the primary limiting factor is the reduction in thrust from the engines.
The higher temperature and lower density of the air results in a lower mass flow through the engine and into the compressor, and thus a lower thrust. The easy answer, of course, would be to turn up the wick and burn more fuel. Indeed, the engine control system does just this when comparing the difference between a cold day and a warmer day. Eventually on a very hot day, however, this has the effect of making combustion components such as nozzle guide vanes and turbine blades a little more pliable than ideal. So, the engine control limits the amount of fuel, and hence the thrust.
According to an old version of "The Jet Engine" by Rolls-Royce, this can result in a 20% reduction in thrust at 45 deg C. I also believe, although somebody else might like to correct me, that the problem is worse on turbojets (such as the Olympus) and low-bypass engines than the high-bypass engines on many other aircraft. This is due to the relative amount of air going through the engine core and compressor compared with the amount bypassing the core.
With limited thrust, you could go for very long runways to reach take-off velocity, or you could unload passengers and bags to reduce the weight of the aircraft, or you could load less fuel. The relatively easy answer is to load less fuel, but this can necessitate a stop at a service station along the way to top up.
Between the level of science understood by this BA spokesman, and the positive spin required in Britain today, the idea of saying that the aircraft could not get off the ground within safety parameters, and therefore took off with less fuel than was needed for the whole flight, is unthinkable.
One of the main design limitations for aircraft engines is the requirement for "hot and high" take-off capability. Hot days and altitude reduce air density, so there are certain airports which engines are designed to cope with to a certain extent, but not necessarily completely. So, you might expect to see reduced loading of aircraft at airports such as Karachi, Mexico City, Denver etc. in certain conditions.