To date, the earliest known reference to giving or placing coins on a corpse or grave comes from ancient Greece. Pausanias wrote that the coins were required as payment to Charon, the mythical "ferryman" who guides the soul across the River Styx to its final destination.
In mediaeval times and later, the practice grew to placing coins on the ground covering a buried body, for the reason sited above (to insure safe-passage for the dead, though likely not for Charon, specifically).
In Burma, during the last half-century, coins were placed in the mouths of the dead to insure the same, safe, passage. Sir James Frazer, the pioneer anthropologist, wrote a piece on the subject in the mid-19th century. You can find more information in Volume 68, Number 7, of "Folk-Lore," (March, 1957, issue).
There is some evidence, though only anecdotal, that the practice also took place in the United States during the "Old West" growth period. Some believe placing coins over the eyelids of a corpse served the same function, though, more practically, this was likely performed to keep the lids closed in rigor.
In Jewish tradition, placing stones on the grave of person shows great respect, love and or admiration. At least as recently as the 19th Century, Jewish families also put coins in the mouth of the departed.
Executive Editor, International Brotherhood of Magicians Internet Portal
Official Historian, and ex-editor, The Psychic Entertainers Association