This is interesting as to what the Rabbis do with the prayers that are placed inside the Western Wall on the Temple Mount.  According to this Jerry Golden article, they eventually bury them in the Kidron Valley between the Eastern Wall of the Temple Mount & The Mount Of Olives.
 
http://www.thegoldenreport.com/asp/jerrysnewsmanager/anmviewer.asp?a=901&z=1