VBA to move cell values to a cell address in each row of a table

Lindsy 

Thanks. The cause of the problem was that there are other data below the update table.

This version should work:

Sub MoveDates()
    Dim ws As Worksheet
    Dim wt As Worksheet
    Dim s As Long
    Dim m As Long
    Application.ScreenUpdating = False
    Set ws = Worksheets("New Template")
    Set wt = Worksheets("Dates")
    m = ws.Range("L31").End(xlUp).Row
    For s = 9 To m
        wt.Range(ws.Range("M" & s).Value).Value = ws.Range("L" & s).Value
    Next s
    ws.Range("L9:L" & m).ClearContents
    Application.ScreenUpdating = True
End Sub