How to improve the speed of VBA macro code?

Just reiterating what has already been said:

Option Explicit

Sub Macro1()

'turn off as much background processes as possible
With Excel.Application
        .ScreenUpdating = False
        .Calculation = Excel.xlCalculationManual
        .EnableEvents = False
End With

    Dim historyWks As Excel.Worksheet
    Dim inputWks As Excel.Worksheet

    Dim nextRow As Long
    Dim oCol As Long

    Dim myCopy As Excel.Range
    Dim myTest As Excel.Range

    Dim lRsp As Long

    Set inputWks = wksPartsDataEntry
    Set historyWks = Sheet11

      'cells to copy from Input sheet - some contain formulas
      Set myCopy = inputWks.Range("OrderEntry2")

      With historyWks
          nextRow = .Cells(.Rows.Count, 1).End(Excel.xlUp).Offset(1, 0).Row
      End With

      With inputWks
          Set myTest = myCopy.Offset(0, 2)

          If Excel.Application.Count(myTest) > 0 Then
              MsgBox "Please fill in all the cells!"
              GoTo QuickExit
          End If
      End With

      With historyWks
          With .Cells(nextRow, 1)
              .Value = Now
              .NumberFormat = "mm/dd/yyyy hh:mm:ss"
          End With
          .Cells(nextRow, 2).Value = Excel.Application.UserName
          oCol = 3
          myCopy.Copy
          .Cells(nextRow, 3).PasteSpecial Paste:=Excel.xlPasteValues, Transpose:=True
          Excel.Application.CutCopyMode = False
      End With

      'clear input cells that contain constants
      With inputWks
        On Error Resume Next
           With myCopy.Cells.SpecialCells(Excel.xlCellTypeConstants)
                .ClearContents
                Excel.Application.Goto .Cells(1) ', Scroll:=True
           End With
        On Error GoTo 0
      End With

    Calculate

QuickExit

With Excel.Application
        .ScreenUpdating = True
        .Calculation = Excel.xlAutomatic
        .EnableEvents = True
End With

End Sub

I’d step through the macro line-by-line to try to locate which line is slow.

Another alternative – although not sure if it’ll speed things up – is to avoid the clipboard and lose the copy/paste so you’d apply a method like the following to move the data:

Option Explicit

Sub WithoutPastespecial()

'WORKING EXAMPLE

Dim firstRange As Range
Dim secondRange As Range

Set firstRange = ThisWorkbook.Worksheets("Cut Sheet").Range("S4:S2000")
With ThisWorkbook.Worksheets("Cutsheets")
    Set secondRange = .Range("A" & .Rows.Count).End(Excel.xlUp).Offset(1)
End With

With firstRange
      Set secondRange = secondRange.Resize(.Rows.Count, .Columns.Count)
End With
secondRange.Value = firstRange.Value

End Sub