Generally speaking, more engines do burn more fuel for a given total thrust requirement. I could not give you a definitive reason why that is so, but according to all the data I've seen, it is true across nearly all engines (of comparable age).
Hard figures for the A340-500HGW and 777-200LR are as follows (the figures for the A346 and 773 are comparable):
Airbus A340-500HGW
engines: Rolls-Royce Trent 556
engine's specific fuel consumption: 0.568 lb. of fuel per lb. thrust per hour
engine's maximum thrust: 56,250
engine's cruise thrust: 11,000 lb.
typical seats: 313
Boeing 777-200LR
engines: General Electric GE90-110B1L
engine's specific fuel consumption: 0.530 lb. of fuel per lb. thrust per hour
engine's maximum thrust: 110,000 lb.
engine's cruise thrust: 19,000 lb.
typical seats: 301
As you can see, the GE90 has about 7% lower SFC (0.530 v 0.568) compared to the Trent 500. This would give the 777 a small advantage alone (the A340 has only 4% more seats than the 777), but the A340's problems are compounded by its higher total thrust requirements, due to its higher weight. Why is it heavier? Primarily because both the wing and the fuselage were pressed into service outside the limits of their original design. The fuselage for example is narrower than that of the 777, so it has to be longer; this increases weight in the same way that bicycle frames with narrow tubes are heavier than those with "fat" tubes (i.e. the walls have to be disproportionally thicker in the narrow tube for equal strength), and also because the longer fuselage must be reinforced to counter bending moments.
The wing's cross section and sweep were likewise optimised for a lighter plane travelling at slower speeds. So the A340 has higher SFC, lower speed (therefore more time in the cruise burning fuel), higher weight resulting in a higher total thrust requirement, and only slightly more seats to counteract these disadvantages. It is no mystery why Boeing sold ten times more 777s than Airbus did A340s last year. The only question that remains is whether Airbus made the right decision in massively reducing development and manufacturing costs by using an existing fuselage cross section. With hindsight it increasingly looks like that was a mistake (because of sharply rising oil prices), and that A340 sales will slow to an inevitable halt in the near future.
On a related note, it looks like Boeing's gamble with the all-new 787 is going to pay off handsomely in the sales battle against the derivative - if highly evolved - A350. Time for Airbus to get some A380s out the door and face the task of developing a new widebody fuselage if you ask me! Fortunately the hot-selling A320 family gives Airbus enough breathing space to get its momentum back, provided we don't see any more harmful turf battles inside Airbus management.