This kind of culture is not unheard of. The most well known (at least to me) is the Fore tribe of Papua New Guinea. As part of their funerary rights they eat the brains of the deceased.
This tradition led to Kuru, an endemic degenerative brain disease.
If this tradition were more wide spread there are ways it could (health wise) work and ways it would end very very poorly.
Disease is certainly a concern, humans are, in many ways...disgusting creatures and death makes us no prettier. Additionally as @TomLeek mentioned diseases in the human to be eaten are obviously communicable to other humans, which isn't nearly as large a concern when eating other animals, exceptions like bird and swine flu as well as ebola not withstanding.
@dsollen, in response to your question in the comments there are some portions that would be relatively safer than others. Muscle tissue and certain organs (the same organs that we eat from other animals, heart and liver for example) would be less inclined to promote disease than the nervous system, the GI tract and the respiratory system.
The kicker here is that human disease spreads in only so many ways, primarily you are talking about those portions mentioned above, blood borne diseases, diseases that are caught like the flu via the respiratory system, and lastly diseases that enter via the GI tract.
Avoiding the respiratory system and GI tract is easy enough but blood borne diseases would be virtually impossible to avoid if you didn't know it was present due to the fact that blood of course permeates every part of the body. And blood borne diseases are generally really not good. The list includes HIV, Hepatitis, Hemorrhagic Fever and while slightly different West Nile Virus and Malaria (these are, as the article mentions) vector based diseases generally caused by insects.
Some general notes to make this tradition safe(r)
- Clean...really really well. Meaning both with soap and like a deer...avoid opening up the internal organs...
- Cook. You've seen the warnings on meat before...cook Bob before you eat him.
- Quick, don't wait a day...don't even wait 12 hours. The sooner it is done the better, decay is rapid.
- Don't eat people that were sick or died of disease.
Cultural traditions that could come from this are potentially there. Your idea of the old being unclean isn't difficult to imagine...when the body stops functioning fully things get gross.
The opposite could be true as well, maybe the young could confer a health bonus or revive those that are older from certain ailments and allowing the dead to live on in those that partake.
Eating hearts could give you courage, eyes wisdom, genitals erm...prowess, and so forth.