Jonathan,

maybe we can convince you to stay with Mathematica and not to use post processing. Rohit's nice solution still can be tweaked simply by:

• using LabelingFunction -> Center;

• avoid writing labels '0' at all;

  bch = BarChart[data, ChartLayout -> "Stacked", Axes -> True, 
     Frame -> True, 
     FrameLabel -> {"Food Age {days}", "Consumption {calories}"}, 
     PlotLabel -> "Food Consumption by Age", 
     ChartLegends -> {"Fresh kill", "Canned food", "Kitchen scraps", 
       "Dry food"}, LabelingFunction -> Center];
  bch /. Text[Style[0, __], __] -> Nothing
  

Addendum:

Here is a somewhat cleaner way giving the same result; just remove the value '0' from your data:

data = {{80, 20, 25, 15}, {40, 60, 30, 20}, {10, 55, 20, 50}, {0, 40, 
     10, 50}, {0, 25, 10, 75}, {0, 10, 0, 100}, {0, 0, 0, 120}} /. 0 -> _;
BarChart[data, ChartLayout -> "Stacked", Axes -> True, Frame -> True, 
 FrameLabel -> {"Food Age {days}", "Consumption {calories}"}, 
 PlotLabel -> "Food Consumption by Age", 
 ChartLegends -> {"Fresh kill", "Canned food", "Kitchen scraps", 
   "Dry food"}, LabelingFunction -> Center]

Thanks, Rohit. Well, no, ChartLabels and friends do not seem to work in this case. I only could make it in this clumsy way (notice the "\n" in the first framelabel):

data = {{80, 20, 25, 15}, {40, 60, 30, 20}, {10, 55, 20, 50}, {0, 40, 
     10, 50}, {0, 25, 10, 75}, {0, 10, 0, 100}, {0, 0, 0, 120}} /. 0 -> _;
bch = BarChart[data, ChartLayout -> "Stacked", Axes -> True, 
   Frame -> True, 
   FrameLabel -> {"\nFood Age {days}", "Consumption {calories}"}, 
   PlotLabel -> "Food Consumption by Age", 
   ChartLegends -> {"Fresh kill", "Canned food", "Kitchen scraps", 
     "Dry food"}, LabelingFunction -> Center, ImageSize -> Large];
dummydata = Range[7];
(* "invisible" barchart: *)
dummybch = BarChart[dummydata, LabelingFunction -> Below, ChartElements -> Graphics[{Opacity[0], Circle[]}]];
Show[bch, dummybch]

Thanks for demonstrating the use of Frame, FrameLabel, and PlotLabel. Guess I'm going to stick with post-processing via Paint.net instead of LabelingFunction, as it looks like there's probably too much artistic license needed for a generic function.