the fridge will become less efficient i would think as the closet heats up.
the way a fridge works is a fluid is allowed to expand in the wantcold area, and when it expands it cools. if it expands enough to be cooler than the air in the fridge, it will absorb heat. then it gets moved to the rear of the fridge and compressed a lot. this heats it up. if it is heated up to a temp higher than ambient, it will dump heat to the air in your kitchen. then it moves back to the inside of the fridge and expands, cools, and grabs some more heat. rinse n repeat.
as the temp goes up over time (due to you putting electrical to mechanical work into the system to drive the compressor), since the door is open, i guess the expanding fluid on the inside will probably have little problem collecting heat, but as the temp rises, the compressed/heated fluid at the rear of the fridge is trying to dump heat to an ever hotter ambient air pool. since heat transfer is directly related to the gradient of temperature, you can see over time the fridge will get worse and worse at dumping heat, and if it doesn't get rid of as much heat, it won't get to as low of a temp when it expands, so the whole deal loses efficiency.
that's my general understanding.