All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad, sallallaahu ‘alayhi wa sallam, is His Slave and Messenger.
According to some scholars, it is likeable for one to bury whatever he removes from his body such as hair, nails, etc., and it is reported that some of the predecessors did that. The reason they mentioned for that is that those things are parts of the human body; and no doubt the placenta belongs to this kind. For further information, see Fatwa 103101.
However, as for as we know, there is no evidence that it is obligatory to bury the placenta, or that a particular Quranic verse is to be recited on burying it.
As regards what you mentioned that whatever is buried will come out as the people leave the spot, it is closer to baseless falsehood and superstitions.
In any case, burying the placenta is the same as burying any part of the body like the nail and others, and there are no particular rites to be done on burying them.Allaah Knows best.