Espero ter entendido sua pergunta. Eu a modifiquei para começar sempre em L2
:
Sub Copy_ATP_Tables()
Dim SectionATP As Long, NextRow As Long
NextRow = Sheets("Results").Range("L" & 2) 'This line defines where to start
For SectionATP = 1 To 35 '36
Sheets("Acceptance Test Procedure").Range("APTSec" & SectionATP).Columns("A:H").Copy _
Destination:=Sheets("Results").Range("L" & NextRow) 'SpecialCells(xlCellTypeVisible)
NextRow = Sheets("Results").Range("L" & Rows.Count).End(xlUp).Row + 1 'Moved to the end of the loop
Next SectionATP
End Sub