I initially hunted old cemeteries when I lived in North Carolina and couldn't think of other places to go. Obviously, it is absolutely mandatory that one get permission to detect there, and you shouldn't carry any big digging tools, or you are asking for trouble. At the time, I only used a hunting knife to dig with.
The REALLY old cemeteries, unless they were churchyards, were often places shunned by the living, particularly here in the Deep South where the cemeteries were the sites of mass burials during yellow fever epidemics. It wasn't until the garden cemetery movement of the later 1800s that they became more parklike, green, landscaped places of beauty. Once the garden cemeteries came along, more living people visited them, and even had picnics and so on there, treating them like parks. Many of the older garden cemeteries are still city parks to this day- such was the case with the one I hunted in my hometown in North Carolina. The manager OK-ed my being there because he knew I wasn't digging deep and I was being respectful and careful. My own great-great grandparents were buried there. I found a number of nice old coins there.
When I moved to coastal Georgia I got the opportunity to hunt a few cemeteries that were much, much older- some dating before the Revolution. But the results in the older, 18th century cemeteries were nowhere near as good as my results in the late-19th century garden cemeteries. Why? I'm sure it is because the older cemeteries got fewer visitors in their day, not to mention the fact that there were more coins in circulation in the late 1800s than there were in the 1700s.
There is certainly a taboo about detecting in cemeteries. In some places I have been, nobody hassled me, but in others, folks gave me some sour looks, including one where I finally had to inform the lady hassling me that I had rock-solid permission to be there. She didn't care- she still thought I was a ghoul. I pointed to her unleashed dog that was pooping on a grave, and shrugged. She gave me another hateful look but finally left. So apparently it was disrespectful of me to dig coins from three or four inches beneath the sod but just fine and dandy for her to let her dogs run and pee and poop everywhere. Don'tcha just love double standards.
Anyway, I have plenty of public and legal property to keep me occupied now, so I seldom if ever feel the need to detect in old cemeteries anymore.