I think the best answer to the question in the title is that a cat’s diseased kidneys do not cause pain themselves but where there is chronic kidney disease there is likely to be uraemia which is a condition in which poisonous waste products accumulate in the body. This is a life-threatening condition and the symptoms of uraemia includes pain. There are many other symptoms of uremia which you can discuss with your veterinarian which includes mouth ulcers (painful) emaciation, bad breath, muscle twitching, diarrhoea and so on. Uraemia is one example. There are other conditions that can be present in a domestic cat at the same time as chronic kidney disease and those conditions can cause pain. I refer to some of them below.
According to Google’s search results (at the date of this post), the author of the top website (wellbeloved.com) pretty well admits that they don’t know whether kidney failure causes pain in cats or not. My careful research using books and the Internet confirms that veterinarians do not normally consider the issue of pain caused by kidney disease per se. This is why you can’t guarantee finding a clean answer on the Internet on veterinary websites to the question in the title. It is surprising that the best book on the market for cat owners to learn about cat illnesses and diagnosis does not refer to “pain” in the index to their book. They have an extensive index. I find this very odd. There may be an historical sidelining of pain as a symptom of illness by veterinarians. I know that at one time, perhaps 100 to 50 or more years ago, many veterinarians did not think that cats felt pain.
Perhaps the better description would be discomfort to varying degrees caused by kidney failure. There are complications such as the fact that feline hyperthyroidism not uncommonly occurs with chronic kidney failure. One does not cause the other but both diseases are common in middle-aged and older cats. Feline hyperthyroidism causes symptoms which are not dissimilar from kidney disease including vomiting, diarrhoea, occasionally difficulty in breathing, weakness and loss of appetite. All of these conditions are going to be unpleasant for a cat. I don’t think you can say that they cause direct pain from an identifiable part of the body but they certainly cause distress and discomfort to varying levels.
Another complication might be that a cat has suffered trauma to the abdomen causing rupture of the bladder or urethra or a fracture to the pelvis. This can cause acute kidney failure. The cat will feel acute pain. The pain is not caused by the kidney failure per se but by the causes of the kidney failure.
Also, infectious diseases especially feline infectious peritonitis and feline leukaemia, nonsteroidal anti-inflammatory drugs and various toxins such as anti-freeze can cause kidney failure. These causes can themselves cause discomfort and pain because of the symptoms arising out of those conditions and diseases.
This leaves us with the task of detecting pain in a cat. Fortunately, I have an excellent page on this website written by people who know all about pain in felines. They are the veterinarians who fight against the declawing cats who were based in Utah, USA. Declawing causes masses of pain. If you click on this link will be taken to a page which lists the signs of pain in a cat.
Postscript: I think that it is useful to ask whether kidney disease causes pain in humans. Humans can express themselves vocally and tell us that they are in pain. The NHS website which is very reliable tells me that pain is not a specific symptom of kidney disease per se although it can cause symptoms such as headaches which are clearly painful. I think that is useful information.
For the sake of completeness, below are some of the signs that a cat is in pain from various sources:
Playing less, more aggressive, changing gait or posture, grooming less, more quiet, hiding, more irritated, changing temperament or mood, sleeping more than usual, a reduced appetite, trembling, agitation, howling, growling, crying, avoids petting or handling, reduced appetite, shallow breathing, rapid breathing, involuntary blinking, tail flicking, avoiding bright lights and being less affectionate towards people.