Jonathan,
Aside from the above mentioned issues (which are dead on) with stress concentration and residual thermal stresses from welding processes, the other reason welding on things which are "touchy" (i.e. boilers, driver centers, etc.) is taken very seriously is because welding can and does actually change the microstructure of the material, since this is a function of the cooling rate of the material. This is actually a whole fascinating field (metallurgy of welding). The fear really is that one of the microstructures you can get from a welding process is martensite (or another a brittle phase) which is a phase which is very hard, brittle, and not safe for use as an engineering material. This is also why during repairs they design an annealing process-- the ultimate goal is to know what the properties are of what you just welded once it's all done.
Regards,
Trevor Hartford