Initially, it does seem like this would be the case. Indeed, if you compare a fridge full of water bottles to an empty fridge, the full fridge will tend to retain a colder temperature inside after you open the door and then shut it.
However, think about it this way. When you open the door of either a full or empty fridge, basically the same amount of air will escape out, and that air will be the same temperature in either case. So both fridges would lose the same amount of coolness (i.e. gain the same amount of heat).
How do you reconcile this with one fridge retaining a colder temperature than the other? The specific heat of water is higher than air, or some might say it has more thermal mass. The energy required to remove heat from each medium is given by the formula Q = C*delta-T, but C is different for each medium. So it can take just as much energy to bring a fridge full of water down by 2 degrees as it would take to lower an empty fridge by 5 degrees, for example.
However, these two scenarios are not quite the same. As someone who pays the A/C bill might realize, the cooler you set the thermostat, the more energy it takes to do that. The energy requirement does not just increase linearly as you lower the temperature. It is more than that. This is because as the refrigerant runs through the cooling coils, it absorbs heat more efficiently from a warmer atmosphere.
In summary, you will have to remove the same amount of heat either way, but the refrigerator compressor may actually run less and use less energy when the fridge is slightly warmer on average -- when it is lacking the water bottles. It is very counter-intuitive.