First, of course, we must exclude accidents, collisions, and component failures -- these obviously can result in injury.
Beyond that, in almost any physical activity it's possible to apply so much force to a muscle that you produce a muscle or tendon pull. In cycling this would be, eg, when you're climbing in too high a gear and exert enormous force on the pedals, sufficient to cause an injury to your leg, arm, or back. This would be more likely in a person who has gained strength in some other activity and only recently begun cycling intensely, however, such that their muscle and tendon strength was imbalanced for cycling -- unlikely to occur in a regular cyclist.
That leaves the knees. While most cyclists never have serious knee problems, a substantial fraction do. This would be due to any or all of three problems:
- Poor knee "geometry", either as an inborn trait or due to some prior injury.
- Cycling with the seat too low and/or in too high of a gear.
- "Imbalance" in the muscles supporting the knee, due to lack of "training" of some of those muscles, while others are being strengthened.
Knee problems can be avoided/controlled by assuring that the seat is at proper height (and that the bike is otherwise properly fit), using a gear appropriate to the conditions and your degree of training/tolerance, and doing knee-strengthening exercises. It is imperative that one not try to "work through" significant knee pain, but instead address it, correcting the problems leading to it.
The most common form of knee pain is "patellofemeral pain syndrome", a condition where the kneecap gets "off-center" relative to the thigh bone.
Finally, it should also be mentioned that there are about a dozen inherited "minor metabolic disorders" -- myoadenylate deaminase deficiency, McArdle's disease, CPTII, and others -- that maybe 3% of the population carries. If a person has been exercising regularly for some time they would already have become aware of these, but if someone begins a program of fairly vigorous exercise for the first time in their life (cycling or othewise) they may encounter muscle pain/soreness that lasts beyond a few days, severe muscle cramps, or other such conditions. If these symptoms continue for more than a week or two they should consult a physician (oddly, the doc to see is a neurologist). In many cases it will be possible to continue reasonably vigorous exercise, if the condition is properly treated and accommodated.