The Tomb of King David is a recent (mediaeval) site which could not be the actual burial place as King David was buried in the City of David which lies further east and south. As it is, the tomb is strangely situated in the same crusader building that houses the Coenaculum.
The constant trickle of Jewish pilgrims to the site increases significantly on the religious festival of Shavu'ot (pentecost), traditionally the anniversary of the death of King David.
1 Kings Chapter 2
10 Then David slept with his fathers, and was buried in the city of David. 11 And the time that David reigned over Israel was forty years; he reigned seven years in Hebron, and thirty-three years in Jerusalem. 12 So Solomon sat upon the throne of David his father; and his kingdom was firmly established.
The Revised Standard Version, (New York: Oxford University Press, Inc.) 1973, 1977.