My dad is 73 and has been on dialysis for 3 years now (hemo). He's been an insulin-dependent diabetic since his mid-40's. My dad has been retaining fluids in his tissues recently, which means removing water is not possible since the fluid is not in his bloodstream. I can't find any information on what happens when/if dialysis stops working. Can it stop working? What kinds of things do dialysis patients usually die from? Any information would be greatly appreciated.

Fluid actually transfers from the tissues to the blood stream during dialysis. That is how it works. Questions you should ask is : Is your dad following the advized fluid restrictions? If not, it is not always possible to remove all of the fluid all at once. Every one is different and tolerate fluid removal differently. Is your fathers blood pressure staying up during treatment? Even if he's following the fluid restriction and not gaining much, if his BP is too low fluid removal is stopped during treatment until the BP comes up.
Does your dad cramp during treatment? This is another reason fluid removal may be stopped during treatment.

Dialysis doesn't 'stop working' so to speak. People die from a variety of reasons, most people come to dialysis with existing problems such as diabetes, high blood pressure, cardiac disease, cancer and the list goes on and on. People pass away for a variety of reasons, usually linked to complications with existing problems.

As long as your dad is managing his diabetes, following the renal diet and fluid restrictrions, and any other regimes the dr has him on, he is doing his best to maintain his health. If his fluid retention is excessive he should definitly speak to his dr.