What exactly does urea do that harms our body? Does it interfere with our cellular processes or disturbs the colloidal onchotic pressure of our blood? I have heard that $NH_3$ damages the brain. What does it actually do?
The denaturing effect of Urea is mostly at very high concentrations. (~Molar levels)!
Physiologically, urea, a product of the breakdown of retired amino acids, will build up rather quickly if it weren't secreted in the urine. Looking into this, it doesn't seem as there is a clear understanding of why this causes a disease, though the buildup of a waste product definitely is a unifying concept.
Uremia - kidney failure - causes a build up of urea in the blood as the kidney eliminates urea from the blood into the bladder. The specific effects of Urea on cell metabolism are probably numerous as the symptoms: anorexia and lethargy, and late symptoms can include decreased mental acuity and coma. Other symptoms include fatigue, nausea, vomiting, cold, bone pain, itch, shortness of breath, and seizures.
Being a metabolic waste product, it certainly would interfere with and inhibit any enzyme that produced urea - the Urea cycle in particular. Its clear that uremia is ultimately associated with cell death, oxidative stress, some cell signalling and proliferation.
I'm just googling around here, but I don't think there is any one single answer as to what goes wrong when the cells are exposed to too much urea.
The toxic effects of ammonia is mainly because of its alkaline nature. Mammals can convert ammonia to urea. Urea is a strong denaturant of proteins. Its chaotropic effects can also affect the cell membrane.