You have a Do Loop that is doing nothing but cycling through the same
cell over and over again.  No need for the Do Loop.  Try this:
Sub Button_Click()
   Dim currentDate As Date
   currentDate = Date
   For Each c In Range("D5:D369")
       If Not c >= currentDate Then
           c.Font.Color = RGB(0, 255, 0)
           c.Locked = True
           c.Font.Bold = True
       End If
   Next c
End Sub

And if you want to exit the sub once you have reached a cell that is
greater than or equal to the currentDate varaible, do something like
below.  The first code I posted will cycle through ALL cells in the
specified range.  Which one you use will depend on how your data is
structured.  If the dates are not in ascending order and there is a
possibility that there is "old" data after the first occurrence of >=
currentDate, then use the first code I posted.  Else, use this code.
Sub Button_Click()
   Dim currentDate As Date
   currentDate = Date
   For Each c In Range("D5:D18")
       If Not c >= currentDate Then
           c.Font.Color = RGB(0, 255, 0)
           c.Locked = True
           c.Font.Bold = True
       Else
           Exit For
       End If
   Next c
End Sub