The most important point to me: If you don't use everything in your fridge before it goes bad, then a full fridge will have costs that a (nearly) empty one doesn't, because you'll be throwing away more food.
If you keep the door closed, then the amount of heat leakage from the fridge is independent of what's in it, so either should be equal; it's (almost) entirely based on the barriers between the inside of the fridge and the outside.
If you open the door, then most of the heat leakage is going to be in the form of convective heat transfer from air leaving the fridge.
At this point a full fridge has an advantage, because more of the air is blocked by the contents of the fridge.
Chad brings up a useful point:
the fridge is likely to work harder to cool things quickly than it is to cool things slowly, even if the heat energy transfer involved is the same.
That said, I think the cooled mass of the fridge itself is so large that it swamps any such effect from the material in the fridge.
MrChrister's point about ice machines requiring heat to dislodge the ice is definitely true, although not all ice machines do things this way.
money.stackexchange.com/users/3513/jprete
Keep the door closed to save money.
And if you have an ice machine, turn that off too.
You need to warm up the tray to get ice out of it and heating something inside your freezer
is expensive.
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Full fridge uses less.
Mostly due to a sort of 'flywheel effect' from the mass inside the fridge.
frankly I think this is more of a science trivia question, If you are trying to save money on power, then there's a lot of other things you can do, and there's tons of websites out there with power conservation tips.
just google it.
A better question here would be what to do with the money you save ;-)
money.stackexchange.com/users/3924/Chuck van der Linden
Well an empty fridge is a waste of 100% of the cooling.
If you are cooling nothing then all of the energy to cool it is wasted.
Air is not very conductive of Heat.
Its why double paned glass is more effiecent than single pane
the air provides an isulation barrier.
So in a fridge you want to quickly bring the temperature of items placed inside down to a storage temperature.
(Around 34F 1c)
Having a fridge full of already cold items will allow for a much quicker transfer of heat out of the item placed within.
And since the heat is dispersed out through a larger mass the amount of cooling required will be the same but because the mean temperature change over time is less the fridge will have to work harder for less time if the fridge is full.
But in the end the amount of energy required to cool a full freeser will be higher than an empty one as there is more heat to disapate.
But the cost of keeping it cold is spread out among many items.
So if you only have 1 orange per day in your fridge and it costs you 30 a month to run your fridge then your oranges have an additional 1 cost on them.
But if you use 20 things a day and it costs you 60 a month it only costs you .1
extra per item.
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In "steady state", the refrigerator only has to pump out the heat that has leaked in from outside.
The amount of heat leaked is related to the difference between interior and exterior temperatures, and the overall insulation quality.
Thus, a refrigerator full of cold food uses the same amount of energy as a refrigerator full of cold air.
In practice, people open the door, take cold food out, put "warm" food in, and the unit may also make ice, dispense cold water, and defrost itself.
Aside from tweaking those uses, (e.g. minimizing time with the door open), a useful trick is to use empty milk cartons or similar containers to fill up unused space.
This helps by reducing the amount of cold air that escapes when the door is opened, and the air in the cartons has low thermal mass, so it cools down rapidly.
When you need the space, you can remove the cartons and set them aside.
Similarly, water-filled cartons may be used as a sort of thermal ballast--leveling out the temperature fluctuations within the refrigerator, particularly if the electric utilities are erratic.
However, they increase the cost of refrigeration when cooling the water in the first place, and when you eventually dump out the water, the expense of cooling it goes down the drain.
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To save money on hydro (electricity), I'd like to make optimum use of my refrigerator.For this purpose, is less energy used when the fridge is full? Or when it is empty?
Saving money on electricity: Is a refrigerator more efficient when empty, or full?
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<cost-cutting><household><electricity>
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A full fridge uses less energy. Make sure you keep the door shut to save money. Keeping the door closed saves money. If you open the door, most leakage is going to be in the form of convective heat transfer from air leaving the fridge. A full fridge has an advantage, because more of the air is blocked by the contents of the fridge and therefore uses less electricity. An empty fridge is a waste of the cooling, because if you are cooling nothing, then all of the energy to cool it is wasted. A useful trick is to use empty milk cartons or water-filled cartons or similar containers to fill up unused space to level out the temperature fluctuations within the refrigerator, particularly if the electric utilities are erratic.  
Keeping the door closed saves money. The amount of heat leakage from the fridge is independent of what's in it, and is mostly based on the barriers between the inside of the fridge and the outside. If you open the door, most of the heat leakage is going to be in the form of convective heat transfer from air leaving the fridge. A full fridge has an advantage, because more of the air is blocked by the contents of the fridge and therefore uses less electricity. An empty fridge is a waste of the cooling, because if you are cooling nothing, then all of the energy to cool it is wasted.   Having a fridge full of already cold items will allow for a much quicker transfer of heat out of the item placed within and since the heat is dispersed out through a larger mass the amount of cooling required will be the same, but because the mean temperature change over time is less, the fridge will have to work harder for less time if the fridge is full. In the end the amount of energy required to cool a full fridge will be higher. A useful trick is to use empty milk cartons or water-filled cartons or similar containers to fill up unused space to level out the temperature fluctuations within the refrigerator, particularly if the electric utilities are erratic.  