I've found this to work very well. It uses the .range property of the .autofilter object, which seems to be a rather obscure, but very handy, feature:
Sub copyfiltered()
' Copies the visible columns
' and the selected rows in an autofilter
'
' Assumes that the filter was previously applied
'
Dim wsIn As Worksheet
Dim wsOut As Worksheet
Set wsIn = Worksheets("Sheet1")
Set wsOut = Worksheets("Sheet2")
' Hide the columns you don't want to copy
wsIn.Range("B:B,D:D").EntireColumn.Hidden = True
'Copy the filtered rows from wsIn and and paste in wsOut
wsIn.AutoFilter.Range.Copy Destination:=wsOut.Range("A1")
End Sub