Now, I can't pretend to have come up with any brilliant theories of my own on this one, but fortunately human knowledge is cumulative. (Discovering calculus remains one of mathematics' greatest moments, and it had to await possibly the greatest intelligence ever to walk this earth: Isaac Newton. Learning calculus, on the other hand, is fairly easy; the high school I attended required us to have mastered the basics even to pass. Imagine that - a genius had to work it out the first time, but after that far more mediocre intellects like us could all benefit...) One of evolutionary theory's living legends is George Williams, and together with Randolph Nesse (a medical doctor) he wrote the stupendously interesting Why We Get Sick. Fortunately for us, the two of them put their minds to exactly the above question, and it's from their book that the following ideas - and quotes - come.
The basic answer is that evolution will not maintain capacities whose average benefits are low. In plain English, this means that even if something is very useful, if it is almost never needed it won't tend to be selected for. On the other end of the spectrum, if an attribute is quite often needed but is associated with heavy costs, it also won't make the cut.
Regeneration of body parts often falls prey to this evolutionary requirement. In many cases (e.g. brain, heart, large body parts), serious damage "was uniformly fatal before the era of modern medicine, and the ability to regenerate these tissues could not be selected for." In other words, if the loss of significant part the body, like the brain or an entire limb, almost always led to death, there would be no opportunity for any later regenerative ability to be selected for. Most people would quickly die whether or not they possessed a tendency to slowly regenerate the tissue, and so "any process that might have allowed our remote ancestors' arms to regenerate has gradually been lost by the accumulation of mutations that have not been selected against."
That is most probably the answer for large and important body parts in which damage led to death often enough. But Williams and Nesse also note that this explanation doesn't hold true of smaller injuries, such as the loss of a finger. In these cases, they offer two further observations.
Firstly, they point out that repairing a smaller deficit often implies that the benefit of doing so would be fairly slight. For instance, you can do quite well with nine fingers - the bonus of having the full array isn't much greater. Furthermore, in many cases these injuries are not often sustained. A combination of these two facts means that, again, there simply isn't a strong enough selection pressure to maintain such an delicate and intricate regenerative mechanism.
Secondly, there is a cost associated with regeneration, namely cancer. One of the best ways of stopping unregulated autonomous cell division (i.e. cancer) is to simply stop all the cells from dividing once they've reached their adult positions and numbers. (An equivalently autocratic decision would be to ban all bicycles as a means of preventing bicycle accidents.) To permit a tissue to regenerate is to simultaneously increase its rate of cancer.
Lastly, it is interesting to note the tissues that do largely regenerate - the skin, the mucosal lining of the gastrointestinal tract, and the liver. Clearly, in these cases, the benefits of regeneration trump the above considerations. The skin is constantly being abraded and damaged from brushes, scrapes and even infections, and so it must quickly regenerate to recover its protective abilities. The lining of the gut and the liver are in constant with the outside world too, in the form of pathogens and toxins. Once again, the eternal attrition and damage necessitate regeneration, a fact that can't be said for the brain or the heart. The liver can grow back if you chop out all but 25% of the organ, but the spleen can't. Perhaps we now know why.