No, you don't get a lot of iron if you eat your placenta.
no, this is not something I had run across in my cross cultural experience: I have no doubt that there is a tribe somewhere that does this, (Wikipedia article) but I haven't worked with any of the tribes mentioned, and indeed, it is hard from the Wikipedia article to tell if this was a common practice or just sometimes done.
In Africa, one of the important jobs of the midwife was to bury the placenta, so it would not be eaten by dogs or hyenas (and would not be unearthed and used for witchcraft spells).
so guess what? It is a fad of the affluent.
YUM! Chocolate covered Placentas!
Placentophagy is an increasingly popular trend in industrialized countries throughout Europe, in Australia and in the U.S.. Proponents of the practice often reference placentophagy's common occurrence among nearly all mammals in nature, and they suggest it offers numerous benefits to human mothers too, including increased energy, improved mood, and more rapid postpartum recovery.
not noted: The reason that animals eat the placenta is to keep the nest clean of odors. They also eat their babies' feces and probably their urine because they lick those areas to encourage the small ones to do their thing.
I should also note that our dogs eat their dead babies too, if they are stillborn. Once I had to remove one from our dog Sophie/s throat because she couldn't swallow it.