You're not closing the excel application properly. You need to close the workbook and quit excel before setting the objects to nothing.
For more information see my answer to Remove Excel Task from Task Manager after running vba.
Public Sub ExportFiles()
Dim X As Object
Dim Y As Object
Dim XL As Object
Dim rs As Recordset
Set X = CreateObject("Excel.Application")
Set Y = X.Workbooks.Open("\\POISE\Data\LCS\DomGroup\ICE NMW Performance and DQ\Tasking MI\2014-15\Production Pack Template\OFFICIAL SENSITIVE Tasking Team Production Pack.xlsx")
Set XL = Y.Sheets("Tasking Records")
Set rs = CurrentDb.OpenRecordset("ALL")
XL.Range("A2").CopyFromRecordset rs
Y.SaveAs "\\POISE\Data\LCS\DomGroup\ICE NMW Performance and DQ\Tasking MI\2014-15\Production Pack Output\" & "Tasking " & "Week " & Format(FiscalWeek, "w") & Format(Date, "yyyymmdd") & ".xlsx"
X.Visible = False
' remove objects in reverse order to creation
Set XL = Nothing
' close workbook before quitting excel
Y.Close
Set Y = Nothing
'close excel before setting it to nothing
X.Quit
Set X = Nothing
rs.Close
Set rs = Nothing