A few thoughts off the top of my head, then I am going to read the other replies. This subject came up a short time ago. Some of the things that were mentioned was about searching in NON BURIAL areas. That might be OK if you have permission. Remember, cemetaries are usually private property. Those, like military cemetaries, that are public property would be considered monuments and would probably be forbidden by the National Parks Service etc. So, I would make sure before going there. I would think, if they forbid photography, for example, in most cemetaries, metal detecting would, probably, not be a desiered activity either. Some people can be very sensitive to such presenses so give some thought.
I believe, It would be wrong to search in the areas where the graves are. Further, some people think that all the items that they find are lost items. That may not be true. Some items are left by mourners as a gift or memory to the one who died. Those would be intended to remain there. It might be a charished childhood toy or a piece of jewelry a military medal... Maybe coins bearing the date of birth or death... Not to rain on someones parade but I saw a picture a few days ago of a charm bracelet that someone found. While that might have been lost by a visitor, it could also have been left as a memorial for a loved one who died. It had charms that seemed to represent zodiac signs, if I recall, and the births of children. I just can't help but wonder if that might not have been left intentionally. Say, by some children, for Mom.
So, Personally, I would stay away from cemetaries. They are a very sensitive area and last thing I would want to do is disturb those who are remeniscing or mourning a friend or relative. Imagine if you were visiting a gravesite and you saw people around the area wandering with metal detectors.
Those are just some of my thoughts, since you ask.