When are multiple hops more efficient than single leg?
In a broad sense, if the flight distance is nearly equal to the max range of the aircraft, the route with shorter legs is prefereable from the point of fuel economy. The reason is simple- more the fuel you carry more the aircraft you have more weight, which requires more fuel.

All other things being equal, the fuel caried is payload- one can see it as the aircraft carrying extra payload or as the fuel eating into the regular payload. Either way, the extra fuel carried impacts on the fuel effeciency of the aircraft negatively.


According to this document, payload burns fuel at the rate of 2.5-5 % of its weight in fuel per flight hour. For the longest non-stop routes in service, the travel time is in the upwards of 15 hours. Taking a value of 3% fuel burn, this means that a significant portion of fuel consumption is actually to carry the fuel itself. This is a significant issue, especially when the oil prices are high.
The increased take off weight will result in more thrust requirement, TO length and time to climb (or more thrust again), adversely affecting the fuel consumption, while increasing engine wear.
While it is true that the longer distance flights spend more time in altitudes with less drag, it is questionable whether that translates into any significant real world benifits. For example, the latest 777 ULR version has a higher aspect ratio to reduce drag, requiring the folding of wings.
Of course, fuel use is only a part of the story. The other significant factor is time- most of the ultra long range flights are targeted towards business flyers, for whom the addition of more than 25% due to shorter legs may be unacceptable. In the end, the deciding factor may be the percentage of seats filled per flight. When SA cancelled the Singapore-Newark flight, one analyst noted:



  “The plane burns a lot of fuel but carries very few passengers,” says Siyi Lim, an analyst at OCBC Investment Research in Singapore. “It didn’t make sense to continue.”



The ultra long distance flights also require additional facilities- extra crew for rotaion, extra food items carried etc. In general, the straight long distance routes have higher fares compared to the ones with stops in between.