I do remember there being a change in whether Excel opens multiple workbooks in the same instance of Excel or not, but I can't remember when this was.
Why are you starting a new instance of Excel anyway?
I've not tested the following but you can try the likes of: 
Sub IMPORT_DATA2()
'Dim oExcel As Excel.Application
Dim strFile As String
Dim FileCorrente As Object
Dim r As Integer
Set FileCorrente = ActiveSheet
'Set oExcel = New Excel.Application
'
mFolder = "E:\Bofetti\CHECK LIST\"
strFile = Dir(mFolder & "*.xlsx")    ' assicurasi che l'estensione del file sia corretta
r = 12    'variabile riga
'inizia ciclo lettura
Do While strFile <> ""
  ' in oExcel ci vanno a finire di volta in volta _
  i file contenuti nella cartella
  Set wb = Workbooks.Open(mFolder & strFile)
    
  FileCorrente.Cells(r, 2) = wb.Worksheets("LGI Certificate Accessories").Cells(7, 4)    'D7
  FileCorrente.Cells(r, 3) = wb.Worksheets("LGI Certificate Accessories").Cells(17, 1)    'A17
  FileCorrente.Cells(r, 4) = wb.Worksheets("LGI Certificate Accessories").Cells(17, 6)    'F17
  FileCorrente.Cells(r, 5) = wb.Worksheets("LGI Certificate Accessories").Cells(31, 4)    'D31
  FileCorrente.Cells(r, 6) = wb.Worksheets("LGI Certificate Accessories").Cells(31, 13)    'M31
  FileCorrente.Cells(r, 7) = wb.Worksheets("LGI Certificate Accessories").Cells(17, 16)    'P17
  FileCorrente.Cells(r, 8) = wb.Worksheets("LGI Certificate Accessories").Cells(18, 19)    'S18
  FileCorrente.Cells(r, 9) = wb.Worksheets("LGI Certificate Accessories").Cells(17, 23)    'W17
  FileCorrente.Cells(r, 10) = wb.Worksheets("LGI Certificate Accessories").Cells(58, 9)    'I58
  wb.Close False
  strFile = Dir
  r = r + 1
Loop
' chiude e azzera variabili
'oExcel.Quit
'Set oExcel = Nothing
Range("A1").Select
End Sub
which can be cut down to (again not tested): 
Sub IMPORT_DATA3()
Dim strFile As String, mFolder, wb
Dim FileCorrente
Dim r As Long
Set FileCorrente = ActiveSheet
'
mFolder = "E:\Bofetti\CHECK LIST\"
strFile = Dir(mFolder & "*.xlsx")    ' assicurasi che l'estensione del file sia corretta
r = 12    'variabile riga
Do While strFile <> ""
  Set wb = Workbooks.Open(mFolder & strFile)
  With wb.Worksheets("LGI Certificate Accessories")
    FileCorrente.Cells(r, 2).Resize(, 9).Value = Array(.Cells(7, 4), .Cells(17, 1), .Cells(17, 6), .Cells(31, 4), .Cells(31, 13), .Cells(17, 16), .Cells(18, 19), .Cells(17, 23), .Cells(58, 9))
  End With
  wb.Close False
  strFile = Dir
  r = r + 1
Loop
Range("A1").Select
End Sub