Let's set the stage here: in the Renaissance, cadavers were not embalmed with chemicals like they are today and were dissected very shortly after death (hence why dissections tended to occur in the winter months when cold weather would keep the cadaver from decomposing as quickly.) For reference, see the bibliography here, which contains some great resources. It would seem that the lack of embalming would lead to greater risk of disease from exposure to decomposing bodies, but that's not necessarily true. As the World Health Organization (WHO) makes clear, there is no inherent risk to handling a deceased body in typical circumstances.

HOWEVER: it IS dangerous to dissect a body that had a contagious illness that could be potentially spread after death (Ebola being a contemporary example.) This is why medical schools and similar institutions today sometimes won't accept body donors who have certain diseases.

In the past: Michelangelo could have contracted an illness by dissecting the body of someone who had a highly infectious disease. This may have occurred if he was sourcing his cadavers from hospitals or almshouses where people were dying of contagious illnesses. But it could be just as likely that he contracted an illness from living people in the same area, and not necessarily from the cadavers he was dissecting. On the other hand, cadavers for dissection in the Renaissance most typically came from public executions of younger, healthy men: thereby reducing the chance they would be spreading disease. On a less dire note, Michelangelo may just have felt a little nauseated or light-headed from the actual act of dissection. Not really a case of "poor health," but a health effect of dissection nonetheless.

So the short answer is: yes, it is possible that Michelangelo became sick from his anatomical specimens, but it would depend on the source of his cadavers. He may also have become ill by contact with sick living people, so it's impossible to say this definitively. Working with cadavers isn't harmful in itself; it depends on the circumstances.