The urate anion originates from purine metabolism, and the hippurate anion is a product of quinic acid metabolism (for which there are several dietary sources). These anions are responsible for some (but not a lot) of the high anion gap metabolic acidosis associated with uraemia.
Urate, uric acid, is the end product of purine metabolism. It is not very water-soluble, and at physiologic pH has a tendency to crystallize, which underlies its tendency to cause gout. It is the end-product of the metabolism of purines, which are ground down to xanthine and hypoxanthine, so that xanthine oxidase can oxidize them into urate. All of this goes on predominantly in the liver. The rest of your body tissues have no way to metabolise urate; they merely soak in it, waiting for the gut or kidneys to excrete it.
Unlike kangaroos and some rodents (who can poop out a solid urate pellet) humans must rely on renal clearance, and about 70-90% of serum urate is cleared by the kidneys. The remainder passively migrates into the gut, where intestinal bacteria feed on it.
There is a weird four-stage process of renal urate handling, and all of the stages occur in the proximal tubule.
In renal failure, the retention of urate begins at the end stage. One can ramp up their excretion of urate quite a lot, and at the end it is the loss of functioning nephrons which frustrates one’s ability to clear urate effectively. However, its buildup does not appear to contribute massively to the uraemic metabolic acidosis. It is merely another organic acid which the dying kidneys fail to clear. Hyperuricaemia is frequently seen as a causative factor in renal failure (eg. high urate causing acute renal failure in tumour lysis syndrome), but it may not be the major reason for the acidosis.
Hippuric acid is a carboxylic acid, and one of the uraemic toxins. In humans, there is a certain normal daily rate of urinary hippurate excretion. Where is coming from? Well some berries and colourful fruits we consume provide us with certain quantities of quinic acid, which gut microflora metabolise into hippurate. According to authoritative sources humans excrete 0.44mg/L of hippurate per day under normal dietary circumstances. Again, this is usually not an issue until the end stages of renal failure. Unless of course one is prone to unusual dietary excesses, or if one is a connoisseur of the volatile solvents.
There is one possible relevance for the hippurate ion which goes largely ignored by the adult intensive care world, and which becomes relevant in the presence of renal failure (or even slightly subnormal renal function). It is the use of sodium benzoate as an ammonium scavenger. Benzoate is given to people with weird urea cycle disorders and hepatic encephalopathy as a new pathway of removing ammonia form the body. Benzoate ends up complexed with coenzyme A to form benzoyl-Coa which is then conjugated with glycine to form hippurate. The dose of sodium benzoate is in the order of 10g/day, which (if al of it is converted) would yield about 70 mmol of hippurate. In the completely anephric patient, this substance would have nowhere to go and a severe metabolic acidosis would develop over days.