Protein and the Kidneys
Don't worry about it if you have healthy kidneys and control your protein intake if you have damaged kidneys. It may be prudent to gradually increase protein intake to higher levels rather than jumping in both feet at a time, but there isn't much on this topic.
It is generally recommended to consume more water during periods when protein intake is being increased. Whether or not this has biological basis is not known, but it may be prudent to do
When looking at active male athletes and measuring urinary creatinine, albumin, and urea no significant changes were seen in dosage ranges of 1.28-2.8g/kg bodyweight. The above study lasted 7 days, but survey research support this lack of association (in post-menopausal women). Although 'high protein' was defined as 1.1+/-0.2g/kg bodyweight, it was associated with better glomerular filtration rate.  The Nurse's study (survey) corroborates these results, but also suggests that this apparent lack of harm does not hold true for renal insufficiency (damage) and that non-dairy animal proteins are associated to damage to a greater degree than other proteins.
There do appear to be functional changes in the kidneys related to protein intake. As protein does modulate renal function, these interactions may lead to damage if imposed acutely onto mice (from 10-15% of the diet, up to 35-45% of the diet immediately) and one study in healthy humans going from 1.2g/kg to 2.4g/kg (doubling) was associated with higher than normal blood values of protein metabolites; a trend was noted for adaptation (increasing GFR) but it was not enough to clear uric acid and BUN over 7 days.
These studies are likely indicative of a 'too much, too fast' sitation, as controlled changes do not lead to adverse changes in kidney function. Thus, it would be prudent to slowly change protein intake over a moderate length of time.
Damaged or unhealthy kidneys
Restricted protein diets are recommended for those with kidney damage, as it slows the seemingly inevitable progression of kidney damage. If protein was not controlled for in those with renal damage, it would accelerate (or at least not reduce) the decline in function.
Protein and the Liver
In healthy persons and rats, there is no evidence to suggest a relatively normal style of protein intake is harmful to the liver when habitually consumed as part of the diet. There is some preliminary evidence, however, that very high protein refeeding after prolonged fasting (>48 hours) may cause acute injuries to the liver.
When is damage seen?
The current standards for treatment of hepatic diseases (cirrhosis) recommend a reduction in protein intake due to the possibility of ammonia build-up in the blood which may contribute to encepalopathy.
At least one animal model suggests that damage may be seen when cycling periods (5 days) of sufficient protein intake, and periods of protein malnutrition. Similar effects were seen after 48 hours of fasting when fed a diet containing 40-50% casein. The latter study noted that the 35% and 50% casein groups had higher AST and ALT levels than the lower protein controls, effectively controlling for refeeding syndroms in general and its adverse effects on liver enzymes. The increases in liver enzymes seen in this study were concurrent with a decrease in the expression of the cytoprotective gene HSP72 and increases of c-Fos and nur77, which are upregulated in response to injury.
Thus, said animal study is some preliminary evidence that high protein refeeding (35-50%) after 48 hour fasting may harm the liver. Shorter fasts were not examined.
Finally, aflatoxin (a toxic mold that is produced from some species of nuts and seeds) is known to be more carcinogenic (cancer producing) when the diet is very high in protein and subsequently less potent in diets lower in protein. This is due to the toxin being bioactivated by the P450 enzyme system, which has its overall activity increased when dietary protein increases. This phenomena also has effects on drugs metabolized by P450, in which the dosage may need to be increased due to faster metabolism.
The above is not an adverse effect of high protein diets per se (as it requires aflatoxin ingestion, which can be avoided) but should otherwise still be noted.
The only other relevant information on the topic is a 1974 study showing that a diet of 35% casein led to increased ALT and AST levels in rats; this study does not seem to have been replicated.
Beyond the above situations, there are no more adverse interactions between dietary protein per se and the liver. It is typically seen as safe to consume protein given you have a healthy liver.
Amino acids are acids, right? What about acidity?
Evidence is theoretically sound, but the acidity of excessive amino acids does not appear to be a clinical concern. It's not potent enough to cause harm to most individuals.
Bone Mineral Density
In looking at large survey research, there appears to be no relation between protein intake and bone fracture risk (indicative of bone health) except for when total calcium intake was below 400mg per 1000kcal daily, although the relation was fairly weak (RR=1.51 when compared against the highest quartile). Other reviews not similar 'lack of correlation despite logic' relations.
One intervention study noted that protein intake was actually positively associated with bone mineral density, but this correlation only was shown when the acidic effects of sulfate (from sulfur amino acids) was controlled for.
Soy protein itself seems to have additional protective effects on bone mass in post-menopausal women, which may be due to the isoflavone content. For more information, please read our FAQ page on Soy Isoflavones.
The role of the Kidneys
Kidneys can acutely increase the Glomerular filtration rate (GFR), or the rate of filtration of the blood. They do this in response to dietary protein intake, and the lack of this compensation in some forms of kidney damage are a reason protein intake is controlled for in kidney disease management.
Additionally, the kidneys serve to regulate acid-base balance in the body via the sodium-bicarbonate buffering system. Disorders in acid:base balance can further pathophysiology (symptoms and signs of the disease) of renal complications.
These protective measures appear to be preserved in healthy kidneys, but begin to fail when they kidneys are otherwise damaged.
The role of Resistance Training
When rats are subjected to an dramatic and acute increase in dietary protein and experience a decline in renal function, resistance training is able to alleviate some of the adverse changes and exert a protective effect.