Have you cross-posted to /r/Archaeology? That might be useful.
I would recommend /r/AskAnthropology. They are very helpful and generally under-used.
Commercial sites in the UK are subject to risk assessments (as are most research excavations run by institutions). This means that risks such as the one you are asking about should be highlighted and appropriate steps taken to lessen or negate the risk. This generally means you have to wear personal protective equipment (PPE). Most of the time this is just Hard Hat and a high visibility vest/shirt, but when things start to get interesting it can get really uncomfortable.
Urban sites tend to be the culprits for the strict PPE that you are asking about. Heavy metals, acidic soils and cess can show up in the pre-ex soil samples which then requires you to wear full hazard suits, safety glasses and gloves as well as what ever else the Health and Safety Executive (in the UK anyway) want you to wear.
Having not worked on a site dealing with plague dead etc I am less sure about the specifics. I think that the general rules involve full PPE (mask, glasses, hard hat, gloves, hazmat suit (generally the lighter kind) etc) when excavating bodies thought to be in contaminated soil. If a lead coffin is found the best bet is to leave it alone, as whatever was inside at internment is most likely still in there and as the wood has rotted away, the structure has gone as well. Things can go horribly wrong and spill/burst if they get moved!
Edit 1: I've not had to wear special PPE to excavate any bodies so far.
Edit 2: [The picture at the top of this article] (http://www.standard.co.uk/news/london/13-skeletons-are-found-in-black-death-pit-under-farringdon--and-there-could-be-50000-more-8535739.html) gives you a bit of an idea. Although it doesn't look like they are wearing anything too special.
Edit 3: Their PPE is related to it being a Cross Rail site and not the presence of plague victims